Dwarves: The entry for dwarves here is mostly just a reiteration of things we have already learned from previous books. Indeed, the race is barely described, and players are directed to the Monster Manual for more information. It's noted that players can use both hill dwarves and mountain dwarves, The differences between the two are negligible: mountain dwarves are a little taller, their skin is lighter, and in the MM they have 1+1 Hit Dice, compared to 1 for hill dwarves. As for PC mountain dwarves no mention is made of that extra hit point, and I'd be inclined to ignore it.
As I understand the rules in OD&D, dwarves could only fighters or fighter/thieves (with clerics and fighter/clerics mentioned as NPCs). In AD&D they have a bit more freedom, as they can now play as fighters, thieves, fighter/thieves, or assassins.
The dwarven resistance to magic is still here, though interestingly it's described as an inherent part of their nature, rather than a cultural aversion. So it's not just that they don't want to be magic-users, it's that there's something in their physical makeup that means they can't be magic-users. In OD&D, this resistance was implemented by treating the dwarf as if he were four levels higher in regards to saving throws vs. magic. In Holmes Basic, it's done by giving dwarves their own line on the saving throw table. Here in AD&D, it's a function of the dwarf's Constitution score, with the character gaining a +1 bonus to relevant saves for every 3.5 points of Con.
Dwarves gain a similar resistance to poisons, though curiously it only applies to "toxic substances ingested or injected". I wonder if this excludes poisons that are inhaled? The definition of ingested could be considered wide enough to cover poison gas, but there is a point of ambiguity there.
The languages known by dwarves are given as dwarven, gnome, goblin, kobold, orcish and common. Previously (in OD&D) they were unable to speak orcish. It's noted that they can't learn more than two extra languages, regardless of their Intelligence scores. It's a bit of an arbitrary restriction, but I suppose it's due to their often insular society.
Their abilities in regards to stonework are pretty much the same as they were in OD&D, but they've been codified in game terms. Also, their chances of success are much greater: in the Basic Set these abilities succeeded about a third of the time; in AD&D the chance of success ranges from 50 to 75%. The ability to determine depth underground is new to AD&D, as far as I can tell.
Dwarves now get +1 to hit against half-orcs, goblins, hobgoblins and orcs. Previously (as per the errata from Supplement I: Greyhawk), this was applied to creatures of the "Giant Class", which was always an ambiguous rule. I tend to play it as applying to all creatures listed under "Giant Types" in the Wandering Monster Tables, which includes all of the above plus kobolds, gnolls, ogres, trolls, giants, gnomes, dwarves, elves and ents. If you use the tables from Supplement III: Eldritch Wizardry the list becomes even larger (though it does include leprechauns, which amuses me for some reason). So yeah, in AD&D this rule has been scaled back significantly. It's probably for the best.
Ogres, trolls, ogre magi, giants and titans all now subtract 4 from their attack rolls against dwarves. In OD&D this was done by having these creatures halve their damage, but the intent of the rule was the same.
(I've checked the Monster Manual, and it's all fairly consistent with what's presented in the PHB. The only major difference is that the dwarven resistances are treated in the MM as they were in OD&D, by saving as if the dwarf were 4 levels above their actual level. It makes sense, as it's unusual for monsters and NPCs to have their ability scores rolled.)