I drove a 2009 C63 for three years before my M3. I like them both. I agree with GIdriver regarding connectedness. The M3 is lighter and feels lighter and handles a little bit better. But the feeling of the handling is much better. I liked the C63 exhaust tremendously. I like the free and high revving of the S65 engine though better. I definitely like the M-DCT over the 09 automatic. The DCT has been the biggest part of the M3's fun for me over the C63.
These cars are really close in terms of overall fun though. I could be quite happy with a 2013 C63 AMG coupe with the performance package.