I'd think that the climates would be about the same. Check out the link. They move the lines around periodically, but Tampa and Melbourne seem to stay in the same zone, as long as you're not too far inland.

Here is a link that might be useful: Florida Hardiness Zones

I think it's about the same zone. 9a. I used to live in Vero Beach. A nice ocean breeze and very shallow water at the beaches. Good luck with your move!

I don't know about summer, but one winter I commuted back and forth from Clearwater to Melbourne each weekend, and it was always 10 degrees warmer in Melbourne.