Achievements to me is like a permanent memory card save on your account of all your accomplishments. I like to keep track of which games I've beaten and whatnot. I won't get EVERY achievement/trophy in the game, just the ones that truly matter to completing the main game. Any other side-achievements are just extra stuff to do if I truly love the game.
Even the Wii U games have their own (hidden) achievement system; collecting stamps that each game offers.