The speed difference will be EXTREMELY dependent upon how good your fitting is for the TT bike. There is no way to overstate the importance of a good fit.
I was accustomed to racing up to 8 hours at a stretch while in the drops on a road bike. Last year, my relay tri teams won two of two overalls while I was riding the road bike -- but I was passed by a few who rode dedicated TT/Tri bikes. That's when I decided to look for something different.
Aero bars -- I never got more than .5 mph benefit compared to riding the same bike in the drops. I suspect you COULD do better, but my positioning on the aero bars was never as powerful or as aero as I would have liked. Aero bars were a non-starter for me.
TT/Tri Bike -- I tried a few bikes and ended up with a BMC TimeMachine02. The biggest difference between it and other TT/Tri bikes was fitting/positioning. With my current fit, I can put out very close to the same power on the TT/Tri bike as I can on the road bike. This is a huge deal. It was true of NO other TT/Tri bike I tried. Coupled with that, my aero positioning allows me to average between 2 and 2.5 mph faster on the TT/Tri bike than on my road bike in the drops. The bike flies. (Granted that the bike itself is more aero and I'm riding Jet Black Plus 6 / Jet Black Disc wheels for a little extra help.)
I'm confident the new bike will help me win a few more relay tri's next year. It's a LOT faster. And, for Strava users. . . when the segments are flattish, the bike is a KOM machine.