Elves: The section on elves begins with a note that all PC are considered to be high elves, the most common sort. The varieties listed in the Monster Manual include aquatic elves, drow, gray elves, and wood elves. All of those will become playable races soon enough, but mercifully we are spared for the moment.
Elves can become fighters, thieves, magic-users, or assassins. They can also multiclass, with the following combinations being legal: fighter/magic-user, fighter/thief, magic-user/thief and fighter/magic-user/thief. This is a big change from the way elves functioned in OD&D, where they were fighter/magic-users, fighter/magic-user/thieves, or thieves. The bizarre way that multiclass elves functioned in OD&D has also been jettisoned; now that system has been properly ironed out, and elves do it just like everyone else.
I didn't note this when I was going through the Monster Manual - and I could very well be incorrect - but as far as I can tell the elvish resistance to sleep and charm spells was introduced in the MM (and appears here unchanged).
The bonus when fighting with bows and swords is not new; it was introduced in Supplement I - Greyhawk. It's clarified here that the bonus doesn't apply to crossbows, and for swords it only applies to short and long swords. They also retain their ability to see in the dark with infravision, to a distance of 60'.
Elves all speak common, elvish, gnome, halfling, goblin, hobgoblin, orcish, and gnoll. Gnome, halfling and goblin weren't on this list on OD&D. Adding goblin makes sense, as I generally assume that goblins and hobgoblins are closely related. As for gnome and halfling, I can say that elves are becoming less insular, and opening the lines of trade and diplomacy with their neighbours.
The elvish ability to sense secret doors has been nerfed somewhat. In OD&D they had a 2-in-6 chance of noticing any secret door they pass close by, and a 4-in-6 chance of locating one if actively searching. In AD&D, a differentiation has been made between secret doors and concealed doors. The elvish chance to automatically notice these now only applies to concealed doors, and has been halved to 1-in-6. If actively searching, they'll find a secret door 2-in-6 times, and a concealed door 4-in-6 times.
Elves get a +1 to Dexterity and a -1 to Constitution. (Am I correct in thinking that these racial modifiers to ability scores are new in AD&D? My notes are becoming a little unwieldy, so it's getting harder to tell, but I'm quite sure I'm correct.)
Elves who are alone and unarmored get a bonus to surprise. This ability was introduced in the Monster Manual, but there it only applied to wilderness areas such as forests and meadows. That limitation doesn't apply here, as opening doors is specifically mentioned as something that negates this surprise bonus.
Gnomes: This is the first appearance of gnomes as a PC race. They can choose to play as fighters, thieves, illusionists, assassins, and can also multiclass. Multiclass characters are said to be able to wear leather armour while using their non-fighter abilities, which brings up the possibility of gnomes spellcasting in armour.
Gnomes get the same magic resistance as dwarves, based on their consitution score. The source of this ability isn't stated here, but it seems unlikely that they're inherently non-magical like dwarves; they can play as illusionists, after all.
(Checking the Monster Manual, I see that the gnomes there get the save bonus vs. poison as well, just like dwarves. That's not mentioned in the PHB.)
Gnomes can speak the following languages: common, dwarvish, gnome, halfling, goblin, and kobold. Like most other PC races, it's their closest allies and their most hated enemies. They can also communicate with burrowing mammals such as moles, badgers and ground squirrels. they have the same limitation as dwarves, and can't learn more than two additional languages.
Also like dwarves, gnomes have 60' infravision, and they can detect certain things when underground: sloping passages, unsafe areas, depth and direction of travel. Where these abilities overlap with dwarves', the gnomes are slightly better.
Again like dwarves (sense a pattern?) gnomes get an attack bonus against certain enemies, and are harder to hit when attacked by others. Their attack bonus applies to kobolds and goblins. Their defensive bonus applies to the same monsters as dwarves', as well as gnolls and bugbears. I guess being a little smaller has its advantages.
Gnomes suffer a bit from being too similar to dwarves, I feel. Their ability to speak with mammals, and the choice of illusionist as a class, is really the only thing that sets them apart. I wasn't at all surprised to see them get cut out of 4th edition, to be honest.