There's not much that we can actually do with spies.. wish there were more missions, as far as we know.
We can rig elections (CS)
We can stage a coup (CS)
We can view enemy city (Civ)
We can steal technologies (Civ)
We can place a spy in our own cities for counter-intelligence (Civ)
We can place a spy in enemy cities for intelligence, intrigue, and as above, to steal techs (Civ)
It seems like a little, but it's a whole lot in the long run. Any information that can help you prepare for an attack/give an advantage is good.
Viewing the enemy city can tell you what they're working on, if they're building WMDs and you need to remove their leader.

The Coup and election rigging can help convert the CS to your side, providing their resources and aid during a war.
Technology stealing will be great if (just an example) your enemy has access to planes but you don't, and you anticipate a war happening soon.