Are these two terms exact synonyms? Or is there some subtle difference between the two?
For example, David Deutsch (2011) writes:
Testability is now generally accepted as the defining characteristic of the scientific method. Popper called it the 'criterion of demarcation' between science and non-science.
Popper (1963) writes:
the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.
It is the difference between existential and universal statements in science. "All electrons in the universe are the same" is not testable but it is falsifiable - a single instance would disprove the assertion. It is important to note that the burden of proof is shifted(!). It is not the party who asserts universality that provides evidence. Popper's idea devloped from the observation that natural science had otherwise to adopt a rather vague conception of "testable in principle" (e.g. the electrons in the universe) which is not different from inductivism.
Popper used testability, falsifiability and refutability interchangeably and presents it as a “criterion of demarcation”. (in “Conjectures and refutations” Page 53, lines 8-14)
He meant by that, a criterion for distinguishing scientific statements, from religious, metaphysical or pseudo-scientific statements.
Testability: The requirement that, any statement/ hypothesis/model/theory which claims to be scientific, should be testable, via empirical observations and if need be experiments.
Refutability and falsifiability:
However, the purpose of the testing is not to verify or confirm the hypothesis but to refute it (refutability) or falsify it (falsifiability).
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Popper adds (in “Conjectures and refutations” Page 48, line 4):” A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice”
Note 1: The “confusion” arises from the fact that the term, ”testability” had been used before Popper in a broader sense, one that stops at “verifiability”. In restating the need for falsifiability, Popper insists that verifiability is not sufficient as criterion of demarcation between science and false science. He is redefining testability and giving it a narrower meaning.
Note2: Popper in my view rediscovered and restated in a clear manner what had been practiced by scientists since at least Newton, and had been proposed implicitly by early modern philosophers (see for example Francis Bacon, in Stanford Encyclopedia of Philosophy (on line), & 5). In a sense, he restored testability to its original meaning i.e. refutability or falsifiability.
Note3: Some scientific theories contain their refutability criterion implicit in their counterintuitive predictions such as the predictions of time dilation, distance contraction or the bending of light by gravity, in the theory of relativity. Any failed prediction refutes, falsifies the theory. In some cases, scientists themselves proposed a refutability criterion for their new hypothesis: I quote Darwin for example (in ‘The origin of Species’, p190) : “If it could be demonstrated, that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down” (end of quote).
A short digression first... Part of the problem we have with this question is that the Philosophy of Science has (historically speaking) over-focused on academic science. That makes a certain amount of sense; the core question of the field is the nature of science, and science as it's commonly understood occurs in academic settings, with academic scientists proposing and arguing over various theories. But in truth, much of 'science' isn't strictly academic. It isn't about people trying to suss out and model fundamental principles; it's more technological, with people trying to use and expand what we already know to produce things. For example, thousands of researchers around the world right now are trying to produce vaccines and treatments for Covid-19. That is not groundbreaking research, we likely won't learn much from the process, but still the struggle to produce these materials is thoroughly scientific.
The point of this digression is that when we restrict ourselves to academic science, 'testing' is at best a synonym for falsification and at worst a completely empty signifier. When an academic scientist tests something, she is not much interested in having the test succeed. a successful test is only useful as:
Tests that fail, on the other hand, are interesting and useful. Academics learn from tests that fail; they create new theories based on tests that fail. This is what led Popper to the 'falsification' model. He recognized that only the failure of a hypothesis had value to an academic scientist.
However, outside of academia, testing has an entirely different meaning; it's an essential part of production. We test plastic toys to make sure they don't poison children; we test materials to make sure they can withstand stresses. Every time we shoot a basketball at a hoop we are testing the theory of gravity (ToG), and every time we start a car we are testing the theory of oxygen combustion (ToOC). If we miss the hoop or if the car doesn't start, we don't automatically think that we've 'falsified' ToG or ToOC. We think we did something wrong, we try to fix the error, and we test it again until we get it right.
So in the broadest perspective testing is distinct from falsification: testing is a process that uses our knowledge to produce and improve outcomes we desire. But when we restrict our perspective to academic science testing collapses into falsification, because the academic world isn't trying to produce anything except valid theories and principles.
Testability can just mean there are tests whose results would increase or decrease one's confidence in a theory, as in Bayesian hypothesis testing, without necessarily requiring that there be any test that would definitively rule a theory out. Whereas Popper grounded his idea of falsificationism in formal logic, namely the idea that any statement involving universal quantification (the ∀ symbol) is refuted by a single counter example. As the Stanford Encyclopedia of Philosophy article on Popper puts it:
In a critical sense, Popper’s theory of demarcation is based upon his perception of the logical asymmetry which holds between verification and falsification: it is logically impossible to conclusively verify a universal proposition by reference to experience (as Hume saw clearly), but a single counter-instance conclusively falsifies the corresponding universal law. In a word, an exception, far from ‘proving’ a rule, conclusively refutes it.
But as various critics have pointed out, in practice one can always come up with supplementary hypotheses as to why a seeming falsification does not actually definitively disprove some general law. For example, historically astronomers noticed irregularities in the motion of Uranus that didn't seem to match the predictions of Newton's theory of gravity, but they realized that the motions might be explainable in Newtonian gravity by positing that Uranus was experiencing the gravitational influence of a new unknown planet in a more distant orbit, and this led directly to the discovery of Neptune.
So, some argue that Popper's falsificationism doesn't really reflect the way science is done to require that there be a possible experimental result that would absolutely falsify a given theory, while agreeing that any scientific theory needs to be testable in the sense of there being tests whose results would strengthen or weaken the theory relative to others.
When Popper says that one is the other, he is overstating his case, exaggerating and being non literal. Confirmability is also testability.
As you have noted, Karl Popper advanced an hypothesis that equated falsifiability with testability. However, an oft-overlooked facet of testability is the logical duality of propositions and proofs. Logical duality says that if there is falsifiability, there must also be a concept of verifiability. Almost no one talks about verifiability independent of falsifiability in philosophical problems, despite its crucial importance to the hard sciences and engineering.
We can prove the non-equivalence of falsifiability and testability with a simple truth table. The columns denote true and untrue claims, respectively, and the rows correspond to whether a claim is verifiable or falsifiable. This means there are four elemental types of claims, as follows:
There are only two types of grey area: Unverifiable truths, and unfalsifiable falsehoods. Importantly there are two quadrants of the chart that correspond to testable and provable claims--not only a one-dimensional category, the "falsifiable", as Popper's fallacy is commonly misrepresented to imply.
As a consequence, falsifiability only matters for claims that are false. If they are not false, there does not need to exist a mechanism for proving them false--that would be immaterial, just as there does not need to exist a mechanism for proving a false claim to be true.
An interesting historical question emerges. Prior to the publication of Popper's opinion, were testability and falsifiability viewed as one and the same? Or must we acknowledge that Popper invented or discovered the true nature of testability? The prior emergence of verification of unfalsifiable claims, including for example that humans can fly, is definitive proof of the recognition of the fact that a true claim needn't be falsifiable in order for it to be verifiable (and consequently, testable).