There are different levels that you can use Compiz. You set it by going to
System, Preferences, Appearance. Go to the Visual Effects tab. There is no
doubt that you need lots of video RAM and a fast processor for Compiz. Your
card should be up to the task, but ATI was slow out of the box to support
the new kernel. It did not have drivers ready by the time 9.04 came out.
Chances are it is using a less than great driver.

As of kernel 9.04 Ubuntu uses a kernel that switches much of the management
to the kernel from xorg. Nvidia had its drivers ready in time for the April
release, but ATI did not. They knew the change was coming but did not meet
the deadline. Ubuntu elected to use the new kernel to keep up with the
competition and take advantage of the newer kernel. This left ATI and Intel
users at a disadvantage.

This situation should be cleared up with 9.10.