bounties are slightly time more consuming and tedious compared to brainless speed rifting.
bounties supply materials that allow your to reforge legendaries (with the intention to get a good ancient piece). the mats are also used to break down items in your cube for item passives (generally for early on in season).
when you join a rift game it puts you into a specific act that supplies a bonus bounty. when you finish all the bounties in that act and talk to Tyrael you get 2x bounty bags.
when you finish all 5 acts in order of the bonus you end up with 10 bags and 10x of each mat which allows you to reforge a legendary twice.