This answer is written with reference to Brahma Sutras with commentary by Adi Shankara (BSSB), translated by Swami Vireshwarananda (1936).
The Brahma Sutras are too cryptic to be understood without a commentary. And Adi Shankara is probably the most reputable commentator of the Brahma Sutras. Adi Shankara to Hinduism is like Nagarjuna to Mahayana Buddhism.
BSSB: This Sutra begins the refutation of the Bauddha school. There are
three principal schools of Buddhism, viz . the Realists, who accept
the reality of both the outside and the inside world, consisting
respectively of external things and thought; the Idealists, who
maintain that thought alone is real; and the Nihilists, who maintain
that everything is void and unreal. But all of them agree that
everything is momentary—nothing lasts beyond a moment.
BS 2.2.18-27 refutes the Buddhist Realists (Theravada?), BS 2.2.28-32 refutes the Buddhist Idealists (Yogachara? Cittamatra?) and Nihilists (Madhyamaka?).
The first misconception is that everything is momentary according to Theravada. Physical matter is not momentary. Only mental states have moments. This is according to the body vs. monkey mind analogy of SN 12.61.
Based on the commentary, the Realists are assumed to believe that everything is composed of mind and matter. The "Bauddha series" refers to the 12 nidanas of Dependent Origination and the aggregates refer to the five aggregates. Nescience means ignorance.
I'll quote a few parts of the commentary.
BSSB: The question now arises, how are these aggregates formed ? Is
there an intelligent principle behind as the cause, the guide, of the
aggregation, or does it take place spontaneously? If there is an
intelligent principle, is it stationary or momentary? If it is
stationary, the Buddhistic doctrine of momentariness is contradicted.
If it is momentary, then we cannot say that it comes into existence
first and then unites the atoms, for that would mean that the cause
lasts for more than one moment. Again, if there is no intelligent
principle as guide, how can the non-intelligent atoms and the Skandhas
aggregate in a systematic way? Moreover, the activity would be
eternal, and there would be no destruction or Pralaya. For ali these
reasons the formation of aggregates cannot be accounted for, and in
their absence there cannot exist the stream of mundane existence.
Consequently, the doctrine of this school of Bauddhas is untenable.
The second misconception is that Buddhism is concerned with metaphysics, ontology and cosmology, just as Hindu schools of philosophy are. The statement above talks about the origin or formation of the five aggregates.
It states that Dependent Origination doesn't explain how the five aggregates originate. If there is an intelligent entity like God, which caused the aggregates to form, then is He permanent (stationary) or momentary? If God is permanent, then the assumption of Buddhist momentariness is violated. If God is momentary, then He might disappear before the five aggregates are put together. If there is no God, then how would the body and the rest of the five aggregates come together? Even if there is literal rebirth, how exactly does rebirth-linking consciousness (if not soul), travel from one place to another and enter the right womb, without the intervention of God? Dependent Origination doesn't explain this.
Buddhism is not interested in metaphysics, ontology and the origin of the cosmos. Buddhism is only interested in solving the problem of suffering (soteriology). The creation or origin of the five aggregates, and whether intelligent design and control is behind it, is not of Buddhism's concern. This is explained by the Parable of the Poisoned Arrow in MN 63 and the discourse on the unconjecturables of AN 4.77.
If one is shot with a poisoned arrow, he should not ask "where does the arrow come from?", "who shot it?", "what is it made of?", "why was it shot?". Instead, it's practical to ask "how to remove the arrow and save my life?"
BSSB: The Sutra can also be explained as follows: The Bauddhas say, if
we hold that the atoms stand in a relation of causality, then no
combining principle of the atoms would be necessary; in that case they
would join of themselves. The latter part of the Sutra refutes this
saying that the causality will explain only the production of the
atoms of the pot of a subsequent moment by the atoms of the pot of a
previous moment, but will not explain the combination of the atoms
into an aggregate, which can take place only if there is an
intelligent agent behind, for otherwise the combination of inert and
momentary atoms cannot be explained.
The third misconception is that Dependent Origination is used to explain the origin or cause of physical matter or even mental aggregates. That's not true. Dependent Origination is used to explain the birth of a mental self-identity, or individuality. It is not used to explain the arising of the five aggregates. A detailed explanation on dependent origination can be found in this answer.
BSSB: The Sutra now refutes that even the successive causality spoken
of the series Nescience, Samskaras, etc. is untenable. Since
everything is momentary, the antecedent thing would already have
ceased to exist at the next moment, when the subsequent thing is
created; so it cannot be the cause of the other. The clay that exists
at the time the pot is created, is alone the cause of the pot, and not
that which existed before and has ceased to exist then. .....
Again on account of the momentariness of things
‘origination’ and ‘destruction’ will be synonymous, for if we say
there is difference between the two, then we shall be forced to say
that the thing lasts at least for more than one moment, and
consequently we shall have to abandon the doctrine of momentariness. ....
If, to avoid the difficulty shown in the previous Sutra, the Bauddhas
say that effects are produced without a cause, then they would
contradict their own proposition that every effect has a cause. If on
the other hand a cause be assumed, then we have to accept that the
cause and effect exist simultaneously at the next moment, i.e. the
cause lasts for more.than one moment, as already shown in the last
Sutra, which would falsify the doctrine of momentariness.
The fourth misconception is that each nidana is the cause of the next one, and since each nidana is momentary, so it may disappear before the next nidana appears. This is wrong, because each nidana is the condition and not the cause of the next one. And all the nidanas (from ignorance to birth) have to line-up at every mind moment, for the birth of the mental self-identity to appear, at every mind moment.
BSSB: The Bauddhas maintain that universal destruction is ever going on, and
that this destruction is of two kinds, conscious and unconscious. The
former depends upon an act of thought, as when a jar is broken by a
man with a stick, while the latter is the natural decay of things. The
Sutra says that either kind of destruction would be impossible, for it
must refer either to the series of momentary existences or to the
single members of that series. The series is continuous and can never
be stopped. Why ? Because the. last momentary existence before such
destruction must be assumed either to produce its effect or not to
produce it. If it does then the series would continue and will not be
destroyed. If it does not produce the effect, the last momentary
existence ceases to be a fact at all, for according to the Bauddhas
existence (Satta) means causal efficiency. Again the non-existence of
the last momentary existence would lead backward to the non-existence
of the previous momentary existence and so on of the whole series.
Again these two kinds of destruction cannot be found in the individual
members of the series also. For owing to the momentary existence of
each member no conscious destruction of it is possible. Neither can it
be unconscious destruction, since the individual member is not
altogether destroyed; for when a pot is destroyed we find the
existence of the clay in the sherds. Even in those cases where it
seems to vanish, as when a drop of water disappears on account of
heat, we can infer that it continues to exist in some other form, viz.
as steam.
Things always being destroyed probably refers to impermanence (anicca).
The statement above says that things cannot be impermanent, because they cannot be destroyed, because the chain of cause and effect of Dependent Origination cannot be stopped, since it keeps going forever. If it has to stop due to destruction, then cause and effect cannot be true.
This argument doesn't make any sense, if we consider all the previous misconceptions above. It's an argument based on multiple misconceptions.
BSSB: Nescience, according to the Bauddhas, is the false idea of
permanency in things momentary. They say that on the destruction of it
Moksha or Freedom is attained. Now this destruction of Nescience must
be one of the two kinds referred to in the last Sutra. If it is a
conscious destruction, depending on the effort of the individual — his
penance and knowledge then this would go(?) counter to the Buddhistic
doctrine of momentariness, according to which Nescience will also be
momentary and cease to exist after a moment of its own accord. And if
we say that the destruction of ignorance is spontaneous, then the
Buddhist instruction ac to the ‘path’ is useless. So in either case
the Bauddha position is untenable.
The idea above is that ignorance (nescience) is momentary, so it appears and disappears. If ignorance is consciously overcome (with effort), then the doctrine of momentariness is not obeyed. If it is spontaneously overcome (without any effort), then the Noble Eightfold Path is useless.
The fifth misconception is that ignorance is momentary and that it disappears completely at every moment (resulting in Nirvana) and then reappears again. This is not right. Ignorance is impermanent, but it's not momentary. Not everything is momentary. Only mental operations and the mental self-identity is momentary. Ignorance and craving can continue to exist through multiple mind moments.
BSSB: According to the Bauddhas, besides the twofold destruction Akasa
or space is a third non-entity. It means the absence in general of any
covering or occupying body. It has been shown in Sutras 22-23 that the
two kinds of destruction are not absolutely devoid of positive
characteristics and so cannot be non-entities. The case of Akasa is
also similar. Just as earth, air, etc. are recognized to be entities
in consequence of their being the substratum of properties like smell
etc., similarly Akasa also on account of its being the substratum of
sound ought to be recognized as an entity. Earth etc. are experienced
through their attributes, and the existence of Akasa also is
experienced through its attribute, sound. Consequently it also must be
an entity.
Hindu scriptures have 5 great elements, and Buddhist suttas usually mention 4 great elements. The missing element is space or akasha.
This is the sixth misconception, because space is mentioned as an element in SN 27.9 and MN 140.
BSSB: If everything is momentary, the experiencer or enjoyer of
something must also be momentary. But that the enjoyer is not
momentary and abides longer is realized from the fact that people have
the memory of past experiences. Memory is possible only in a person
who has previously experienced it, for what is experienced by one man
is not remembered by another. So the agent of the experience and the
remembrance being the same, he is connected with at least two
moments—which refutes the doctrine of momentariness.
The argument to support an individual soul is that personality and memory exists continuously, so it must depend on the existence of an individual soul for its continuity.
This argument can be easily refuted. A person who suffers from brain injury or Alzheimer's disease, or the effects of drugs, could forget his personal identity or have his personality changed.
Also, computers remembering data or configuration using data storage, does not prove that computers have an individual soul.
BSSB: The Bauddhas say that from anything that is eternal and
non-changing no effects can be produced; for that which does not
change cannot give rise to effects. So they say that the cause
undergoes destruction before the effect is produced. The seed
undergoes destruction, and then the sprout comes out. In other words,
existence springs from non-existence. The Sutra refutes this by saying
that if it were so, then the assumption of special causes would be
meaningless. Anything might spring from anything; for non-entity is
the same in all cases.
This is answered by SN 12.17. Future consequences are neither completely unrelated to past actions (annihilationism i.e. no one owns it), nor are they completely related to past actions (eternalism i.e. the same individual soul owns it). The Buddha takes the middle way through explaining this using Dependent Origination.
BSSB: From this Sutra begins the refutation of the Idealists among the
Bauddhas, according to whom only ideas exist and nothing else.
According to them the external world is nonexistent.
This is the seventh misconception. Yogachara doesn't say that the external world doesn't exist with respect to metaphysics, ontology and cosmology. Rather, it says that, based on soteriology. This is explained in this answer.
It's a method or tool to solve the problem of suffering. It's not an explanation of how the universe exists. This is the second misconception, applied to Yogachara.
BSSB: Nihilism of the Bauddhas goes counter to everything. It goes
against the Sruti, the Smriti, perception, inference, and every other
means of right knowledge and so has to be entirely disregarded by
those who are mindful of their welfare.
The eighth misconception is that Madhyamaka states that nothing exists at all, with respect to metaphysics, ontology and cosmology. That's not true. Rather, it says that, based on soteriology. This is explained in this question and answer.
It's a method or tool to solve the problem of suffering. It's not an explanation of how the universe exists. This is the second misconception, applied to Madhyamaka.
In conclusion, the Brahma Sutras have a mostly misconceived or distorted basis of Buddhist doctrines, upon which refutations are made. Some refutations can be easily dismissed using modern day knowledge of science.
This (distortion) agrees with the conclusion of Gregory J. Darling's book "An Evaluation of the Vedantic Critique of Buddhism", that was quoted in Andriy's answer.