Major things:
1. Terminology changes. Permanents are "on the battlefield" instead of being "in play", and "enter the battlefield" instead of "come into play". Spells are "cast" and abilities are "activated" instead of being "played" (land are still played, and "cast" is a subset of "play"). Cards are "exiled" instead of "removed from the game".
2. Mana burn is gone. Unspent mana just goes to waste now, instead of dealing damage to you.
3. Combat damage no longer uses the stack, so you can't have a Mogg Fanatic lethally block a 2/1 creature and still sacrifice it to deal 1 damage to something else. It has to be one or the other.
And later on, they changed it so that legendary permanents only look at your side of the board, so both players can have Akroma out. If you cast Akroma again, then you can now choose which one to sacrifice and which one to keep, as long as you just have one on the battlefield in the end.