The damage is weapon damage; whatever the weapon does for damage, you still do that damage, plus additional damage of the same type.
As proof, I will use the English dictionary:
extra: a) more than is due, usual, or necessary : ADDITIONAL
D&D is not a game full of "lawyerese." RAI, it is presumed that all the normal rules of English apply, unless the rules explicitly state otherwise. Once you start reading rules as if they were written in English, and not a computer language, you'll see this class feature isn't ambiguous at all.
If you're not fully convinced, try this exercise:
You hand your friend a napkin.
Your Friend: Thanks, can I get a few extra?
What would you do in this situation, assuming you wished to accommodate your friend?
A: Hand them 2-4 more napkins.
B: Hand them 2-4 packets of mustard.
The RAW is depending on the ordinary use of the word extra to tell readers what kind of damage it is. As a DM, and as a player for other DMs, I've always seen the word "extra" used in the normal sense of the word, unless explicitly stated otherwise. For example, flaming burst weapons explicitly state that they deal fire damage, such as "... an extra 1d10 points of fire damage ...". Short of any words in the book or errata that contradict the ordinary meaning of English as we use it today, we must assume that the normal usage was intended.
This is also further supported by the apparent "fluff" in the rules:
Beginning at 1st level, you know how to strike subtly and exploit a foe's distraction.
But this isn't fluff. This is telling you how the ability works. The rogue looks for an opening to hit a weak spot, such as armpits, gonads, etc, and then expertly guides the strike to that location. The rules aren't more explicit than just "general weak spot," but is basically saying it's a "called shot."
The weapon is still doing all the damage, but it's hitting a particularly weaker spot, causing more damage than normal. This is a common trope (warning: TV tropes) in games, and the rogue is the epitome of such a combat fighter that takes advantages of distractions to take down opponents quickly.
Sneak Attack relies entirely on the weapon you're using. The weapon
sets the damage type of the extra damage, and the weapon determines
whether you can Sneak Attack at all; the weapon must be in the ranged
category or have the finesse property. #DnD
If the weapon is setting the damage type, then surely it must be the source of the damage. That must be, otherwise we're saying that the rogue has some Supernatural Ability to turn their hands or something into blades or be able to cast spells that mimic a weapon's damage, neither of which are true. Occam's Razor applies here. The most simple answer is the correct one.