Sneak attack says:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon.

Would the extra damage from sneak attack bypass the resistance of a creature that had damage resistance/immunity to damage from weapons?

Damage Immunities: Bludgeoning, piercing, and slashing from nonmagical weapons

One aspect I am specifically curious about is that sneak attack does not say that it is weapon damage (nor does it say it isn't); all it says is that it is "extra damage". Is sneak attack considered to be damage from a weapon or is it something separate?

For example, consider the case of a rogue with a nonmagical dagger hitting a creature with the above damage immunity. Obviously the dagger does not do any damage, but what about the sneak attack damage? What about when the creature only has resistance?

Resistance/Immunity would apply (because the rule used in the question is defunct)

It's debatable whether or not sneak attack's damage is from the weapon: but one thing I can be sure of is that sneak attack damage is from the attack. This can be seen in the PHB (p. 196, bold added), where it states:

When you score a critical hit, you get to roll extra dice for the attack’s damage against the target... If the attack involves other damage dice, such as from the rogue’s Sneak Attack feature, you roll those dice twice as well.

That's very important, because the rule you're quoting about "nonmagical weapons" is now defunct. See the Monster Manual Errata which states:

Damage Resistances/Immunities. Throughout the book, instances of “nonmagical weapons” in Damage Resistances/Immunities entries have been replaced with “nonmagical attacks.”

The same errata goes further, explaining what magical attacks actually are (and implies what they aren't):

Particular creatures are even resistant or immune to damage from nonmagical attacks (a magical attack is an attack delivered by a spell, a magic item, or another magical source). (MM, p. 8 post errata)

And rules designers have explicitly stated that sneak attack is intended to do the same type of damage as the weapon used in the attack. Jeremy Crawford has stated:

Sneak Attack damage is meant to be of the type that's dealt by the weapon, and it's subject to resistance.

So although it may be debatable whether or not sneak attack damage comes from the rogue's weapon, it definitely is part of an attack and will do "bludgeoning, piercing, or slashing" damage if that's the type of damage done by your weapon. And so, if the weapon used in this attack is a nonmagical dagger, sneak attack is part of a nonmagical attack and would deal piercing damage, and thus the resistance/immunity
to "piercing... from a nonmagical attack" would apply.

Yes, according to Jeremy Crawford in this tweet:

Twitter User: sneak attack has the same damage type of the weapon used?

Jeremy Crawford: That's correct.

The attack damage is the same type as the weapon. If the sneak attacker used a magic weapon, the additional damage would be magic damage of the same type of damage as the weapon and would would not be resistant (the condition of being non-magical would not apply). If the attacker used a non-magical weapon, the resistanceimmunity would apply.

In the example written above, a rogue with a nonmagical dagger hitting a creature with the above damage immunity (Bludgeoning, piercing, and slashing from nonmagical weapons) the sneak damage is also non-magical piercing damage and the immunity will apply. Likewise, the resistance to non-magical piercing damage would reduce the damage by half.

You still get to add Sneak Attack, but it might not matter anyway.

As MarkTO pointed out, the prerequisites for Sneak Attack are in the type of weapon used, not the kind of damage it deals.

However, as stated here, Sneak Attack damage is of the same type as the weapon used to deliver it. So if the weapon’s damage is already affected by the target’s resistance or immunity, Sneak Attack damage will be similarly affected. If not, then not.

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.

JC says:

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.

It doesn't matter in this specific case

As Gandalfmeansme addresses in his answer, the nonmagical weapons bit has been replaced with nonmagical attacks (which Sneak Attack usually is). This removes the ambiguity of this particular inquiry. However, in case it comes up in connection to another feature:

RAW is unclear on whether it is damage from a weapon

Whether or not the Sneak Attack damage is considered weapon damage is ambiguous. There is no source of the damage listed, and even the damage type is only implied (and reinforced by the lead rules designer, Jeremy Crawford). While it may be inferred that it is from the weapon, there is insufficient evidence to this claim in the RAW.

Because of the ambiguity in the rules you'll have to ask your GM how it will work at your table.

Some possible rulings...

... and my issues with them:

1. The damage type proves the damage source
   
   
   • The status of the damage type as the damage source (i.e. the same as the weapon) is not to say the damage source is the same. While I can see where this assumption comes from, it lacks hard support.
   
   


2. The requirement of a weapon proves the damage source
   
   
   • Just because the feature required a finesse or ranged weapon, doesn't mean the damage itself is "from a weapon." There is a counterexample involving sneak attack and magic stone where the sling attack is a ranged spell attack.