With a recent hotfix, Rifts now have +25% Legendary drop rate so there is another huge incentive to going on Rift runs.
One benefit of doing bounties/Nephalem Rifts is the blood shards that you gamble with at Kadala. The Rift bosses drop pretty large chunks of them at a time so it makes it much easier to scratch the gambling itch.
Also, I'm not sure if it is due to Adventure Mode or because the Nephalem Rift itself, but mob density also seems to be a lot higher than your average area. The high density, of course, results in more experience/drops. There are a large number of champion and elite packs spread throughout the rift that you can continue to kill after the rift boss is dead as long as you don't speak to the Nephalem Spirit in town.
There are some unique aspects to the rifts themselves there are unique pylons that can greatly speed up your monster killing and looting:
- Shield - Impervious to damage for 30 seconds.
- Conduit - Lightning zaps your surroundings for 30 seconds.
- Power - All attacks do +400% damage for 30 seconds.
- Speed - Maximum run speed greatly increased for 30 seconds.
- Channeling - All resource costs and cooldowns are reduced
Finally, actually going to the rifts is the only way you will be able to unlock the Nephalem Rift achievements though as with all achievements your mileage can vary based on how you rank achievements.