In DnD 3.x and related systems, you threaten a critical hit on a natural 20, but then have to confirm the crit by making another to-hit roll. If your followup roll beats the target's AC (or is another natural 20), then the hit was a critical hit and you make another damage roll.
There are plenty of variations to this, such as weapons, feats, items, stances, and spells that increase the "threat range" to be 19-20 or even 15-20 (or larger) if you stack things cleverly, and similar enhancements to cause more than one extra damage roll, and chain extra stuff off the crit.
I'm only asking about the simplest case here. From an abstract point of view, scoring a crit really just means "make another single attack", right? The attack and damage roll are the same. Am I missing something here? Are the two concepts really equivalent (before you get into all the add-ons that make crits more common or more damaging)?