No to combat. It would be certainly neat if it did though.
Partial Yes to spawns now, some special mobs only appear during certain weather conditions (iirc, its only Odin and his lightning weather atm, but I seem to remember something requiring the gloom weather).
It does also affect fishing, even more so with the highend fish (stupid raincaller), and greatly so with the recently added "Big Fish." The Sightseeing Log also usually requires that you visit each vista during a specific type of weather and time-of-day.
The 2.35 update (Aug 19, 2014) added "Skywatcher" NPCs that will give you an in-game weather forecast for the next 8, 16, and 24 hours (Eorzea time).