A critical hit and a 20 on the attack roll are two different things.
The sword's description states (emphasis mine):
When you attack a creature that has at least one head with this weapon and roll a 20 on the attack roll, you cut off one of the creature's head.
Beware: it does not say a 20 on the d20, but a 20 on the attack roll. Per PHB, an attack roll is a d20 roll plus modifiers (bold & italic mine):
Ability checks, attack rolls, and saving throws are the three main kinds of d20 rolls, forming the core of the rules of the game. All three follow these simple steps.
1. Roll the die and add a modifier. Roll a d20 and add the relevant modifier. This is typically the modifier derived from one of the six ability scores, and it sometimes includes a proficiency bonus to reflect a character’s particular skill. [...]
2. Apply circumstantial bonuses and penalties. A class feature, a spell, a particular circumstance, or some other effect might give a bonus or penalty to the check.
[...]
This is confirmed also on the Attack Rolls section in the PHB:
To make an attack roll, roll a d20 and add the appropriate modifiers.
Per the rules as written, if a character with a vorpal sword has a bonus to hit equal to +8, then rolling a 12 on the d20 provides an attack roll of 20, triggering hence the sword's power. But it is not a critical hit.
There are some magic items (weapons) that explicitly require a critical hit and not a 20 on the attack roll, for example the nine lives stealer.
Adamantine Armor does not protect from vorpal sword's effect.
As stated in the previous posted answer, Adamantine Armor protects from critical hits, which are not suitable triggers of the vorpal weapon: the target of the attack still loses theirs head in case of a 20 on the attack roll.
How does it affect the probability to trigger the sword's effect?
Consider the example above for a generic character: with a bonus of +8 a 12 on the d20 gives an attack roll of 20. The probability to get this outcome is 1/20. The probability of getting a critical hit is 1/20. The probabilities are the same, thus adopting the "critical rule" for the vorpal sword does not affect the probability.
There are differences when some bonuses can be applied to the attack roll, for example the 1d4 from a Bless spell (see point 2 in the definition of d2 rolls). In this case, under the same +8 base bonus, the probability to have a 20 on the attack roll is1 1/10.
Moreover, there are some subclasses (Champion, Oath of Devotion Paladin) that benefit from improved range for critical hits: the DM has to decide how to treat this subclasses under using crit hits for triggering the head-severing effect (indeed, it doubles the probability for these subclasses).
1 This is given by
$$
\begin{eqnarray*}
P(20|\text{Bless}) &=& P(8 \text{ on d}20)\cdot P(4 \text{ on d}4) +\\ &&P(9 \text{ on d}20)\cdot P(3 \text{ on d}4) +\\
&&P(10 \text{ on d}20)\cdot P(2 \text{ on d}4) +\\
&&P(11 \text{ on d}20)\cdot P(1 \text{ on d}4) +\\
&&P(12 \text{ on d}20)\\
&=& \left(\frac{1}{20}\cdot\frac14\right)4 + \frac{1}{20}\\
&=& \frac{1}{20} + \frac{1}{20} = \frac{1}{10}\\
\end{eqnarray*}
$$
