Compared to, for example, completing 5 bounties in each of 5 different games.
The answer used to be no, but Patch 2.3 has changed this.
When you start a game in Adventure Mode, a single act will be marked as a Bonus Act. Completing all bounties in that act will award you with a Bonus Cache, similar to the Horadric Cache you normally get. The Bonus Cache will contain gold, blood shards, and crafting materials, including the new act-specific crafting materials that are required to craft set and legendary pieces.
Once you acquire this Bonus Cache, the Bonus Act changes to one of the other acts in the game. So by completing all of the bounties in a game, you will obtain the extra gold, blood shards, and act-specific crafting materials - as long as you do the acts in the order they are marked as Bonus.
You could get all this by making five separate games, but there is no guarantee the first Bonus Act will be a different act in each game. By completing all bounties in a single game, you can guarantee you receive the extra act-specific crafting mats for all acts.