With warrior guild's passive, you deal more damage to undead. Undead are vulnerable to fire, sometimes they fall when they are it by fire attack, but the bonus damage is very low or inexistent.
There is some resistance, try a bow against dwemer for example. But they are very low, for example fire-based attack against all enemies in the City of Ash are better than other type because of the flame dot.

This is easy to test, make fire/ice/lightning staffs of the same level and quality. Pick a trait that doesnt affect the damage or non at all.

Now walk up to a fire/ice/storm atronach and see how much damage you do with normal fully charges heavy attacks. Be sure to test it on similar level enemies, south of reapers march should be good to test this.

What you probably will see, less damage if you use a ice staff on a ice atronach and more damage if you use a fire staff for example. I think the difference in normal attack damage will be 5-10% but Im curious about the test results.

Pretty sure vampires are weak to shock damage (i.e. the last boss in Spindleclutch...or at least he is)

So, since I am utterly unable to do any kind of data-mining, I simply crafted four different staves and I used my damage-tracker add-on (FTC). I tested several kinds of mobs with light attacks, and I recorded the damage I did to each. Obviously I only tried a few mobs here and there, I couldn't do an extremely thorough testing. Here is what I found:

The differences between different staves are rather small, but they are there.

In every case, the Restoration staff does the least amount of damage.

Apart from Restoration staves, most mobs seem to follow a general rule: ice and fire do the same amount of damage, shock does a bit more damage than ice and fire. This rule applies also to some mobs which, from a purely lore-based point of view, should have different weaknesses and resistances. For example: a Storm Atronach takes more damage from a lightining staff than from a fire staff (even though Storm Atronachs are supposed to be resistant to shock); a Frost Troll also takes more damage from a lightning staff than from a fire staff (even though trolls are supposed to be weak to fire). The rule applies to many daedra as well, for example clannfears and dremora. In this case it is lore-appropriate insofar as it takes into account these daedra's weakness to shock, but it doesn't take into account their resistance to fire.

Fire and Frost Atronachs seem to be lore-appropriate. From highest to lowest damage, they go as follows:
Fire Atronach: Frost, Shock, Fire, Restoration
Frost Atronach: Fire, Shock, Frost, Restoration
I can't quite understand why, but Storm Atronachs on the other hand follow the general rule: Shock, Fire and Frost, Restoration.

Undead follow some rather strange patterns, again not very lore appropriate. Skeletons, draugr and humanoid ghosts follow the general rule, so they take more damage from shock, and equal damage from ice and fire (thus contradicting the lore). Zombies and wraiths, on the other hand, take the most damage from fire, followed by shock and ice. Restoration ranks last in every case, even against the undead.

Spriggans also seem to be lore appropriate, as they take more damage from fire than from shock.

Spriggans and zombies also take "explosion" damage from fire attacks, but I didn't see it for any other mobs (maybe I just wasn't paying enough attention).

In conclusion: there seems to be a general ranking of damage types which is followed for most mobs. Some mobs divert from this general ranking in ways which are consistent with the lore. Other mobs follow this ranking anyway, even in cases where the lore would dictate otherwise. So weaknesses and resistances are lore-appropriate in some cases only.

Obviously it should be pointed out that I only tested this on a few mobs in some specific areas. It may well be that a storm atronach in Auridon behaves differently from a storm atronach in Craglorn, but I have not tested that.

I would love to test this fully but consoles have no add-ons, at least at the moment...sigh* Undead, spriggans, vampires, and even mobs with fur seem to be weak to fire but I really don't know about daedra