/SLAP You BEEN SLAPPED
Ok, so the primary difference is when you are allowed to attack. Both PVP and PVE allow PVP, to begin with.
In PVP:
You can attack anyone who is the enemy in your land. You cannot be attacked ikn your friendly land unless you engage the enemy first.
You can attack anyone who is the enemy in contested lands and you can be attacked by the enemy in contested lands.
You can only attack those who are engaged or have engaged in fighting in enemy lands, but you can be attacked by anyone just for being there.
In PVE:
You become what is called 'Flagged' which means you can now be attacked. You become flagged if you attack an enemy (of course he has to be flagged already), if you attack an opposing force NPC or if you aid a player engaged in PVP. Yes, healing a PVP player while he is engaged flags you (this makes perfect sense...otherwise a bunch of healers could remain untouchable and constantly heal the warriors fighting). Oh, and entering an opposing controlled city auto flags you.
You can also flag yourself by typing /PVP. The flagging is temporary and in a fairly short time you are back to neutral.
So on a PVE server you could run right through the middle of a battle in a contested land and until you did something to get yourself flagged, remain untouched. On a PVP server you would be inherently a target.
There is more to it, but that is just about it in a nutshell.
Read this page: http://www.worldofwarcraft.com/info/basics/realmtypes.html