When I got my road bike, I was too fat to ride on the hoods and breath. So I rode it like a flat bar setup. Once a lost enough weight to comfortably ride in the drops I also added clip on aero bars. My experience with speed from fastest to slowest with the same perceived effort is:
aero bars
drops
hoods
tops (like a flat bar)
I think the advantage is mostly aero, however I can ride noticeably faster for a given heart rate (I don't have a power meter) when on the aero bars. I've been told it's due to hip angle. I run a fast forward seat post also. At first I only used this seat post for the aero bars. But I found I'm more comfortable with it regardless of position so I leave it on.
Good luck!