What's the benefits of doing Adventure Mode/Bounties compared to Rifts?

Solution 1:

One of the main reasons people do bounties (besides getting keys to enter rifts) is certain legendaries can only be acquired from Horadric Caches, which you receive upon completing all 5 bounties in one act. Some of these legendaries are very important for end game. For example, Ring of Royal Grandeur is a must for most, if not all, end game builds.
T6 (Torment 6) Bounties now guarantee an act legendary (Thanks to @Frank for this info).

The full list of items that drop only from Horadric Caches can be seen here.

Solution 2:

There are a few tradeoffs. As mentioned by Chippies, you need keystones from bounties and obtaining the Royal Ring of Grandeur is a must and is considered BiS (Best in Slot) for almost every build being used. Aside from farming for the ring though, there is still reason to do the bounties once you have it.

Rifts are great for loot. There are many elite packs and the end boss drops a good set of loot as well. This is a good way to get materials for crafting. Rifts are also a good way to practice builds for greater rifting. What rifts tend not to be is fast. Even with a full group, the rift is usually done in order.

Bounties are great for quick experience. The proof here is that basically every seasonal character leveled taking advantage of a few bounties which were easy to complete (namely Miner's Gold). When in a group, the bounties may be done simultaneously (known as split bounties) and that makes them very fast to do. Essentially the time it takes one person to do the slowest one is how long the entire set of bounties take. Factor in that now on T6 you get a 100% legendary drop rate from the horadric cache and these start to become very appealing.

In the end, it is just another option to have fun. I like that they are giving players different avenues of progression and this is just one of those places where if you get tired of doing rifts, you can switch over to bounties. Tired of both? Try a greater rift :) None of that works? Try your luck with the goblin portal.

All in all, they are not mutually exclusive and each has their own unique advantages.