The end game is really just finding different gear and improving stats on gear.

Adventure mode allows you to do Bounties, which are like quests, without having to go through any of the story cuts. Also, you can do Rifts, which are timed random dungeons that have a greater chance of legendary drops. Greater Rifts also give legendary gems, which are very powerful and can be leveled up by doing higher levels of Greater Rifts. So, once you have gone through the campaign once, there is really no reason to go back to it and Adventure mode offers an alternative way of playing.

I haven't played the console versions so I don't know if there are any major differences from the PC version.