Itemization is also a factor, and since other people already covered most of the difference in champions (SO to EndlessSorcerer), so I figured I'd hit items real quick. Here's a quick example of a tank, and a bruiser (I won't cover the subsets tho, itemization differs between classes but Ima keep this simple) item.
is a tank item. Its main purpose is to lower enemy dps not through stats, but actually directly lower the enemy's dps. It rewards you being within range as well and disrupting the enemy team. Less enemy dps means not just you, but your entire team is taking less damage via autoattacks.
is a bruiser item. It offers both defensive (to stay alive in a fight, bruisers aren't [usually] assassins) and offensive (to deal more damage, derp) stats. The passives are designed to help you deal even more damage in a fight, and is clearly intended to be a primarily offensive item that helps and rewards staying in a fight longer (like Frozen heart) with a different reward.
There's a few items that overlap a bit (randuin's, RG, etc) depending on the situation, similar to how some carries may choose a bloodthirster over BORK if they need the defensive passive.