I'm at the War Table and I want to perform a certain mission, but there are a number of advisers to send. Does it matter which one I choose? Will I get different rewards if I send Leliana instead of Cullen, does Josephine have a higher rate of success on certain missions?
There are some operations that can lead to long chains of additional operations if you use some advisors, but will just cut off if you use others.
For example, the Sutherland operation chain can be cut off if you use the wrong advisor.
If you use the correct advisors, you can get a very long chain of operations (and quests) that develop a team of adventurers for the Inquisition. Sutherland is located in the inn in Skyhold, and if you send him on operations, he ends up recruiting and gearing up a team. He can also get his team captured during an operation, leading to a quest where you need to go rescue him. If you use the wrong advisor, he just gets sent to the kitchen and the whole chain ends there.