Short story long, even though the 4 series is the best supported by the closed driver and tremendous strides have been taken with the two open source drivers, ATI support for desktop Linux (video acceleration and 3d) ranges from unsupported or unusable to barely functional with caveats. The most optimistic estimates are at least six more months until most ATI hardware reaches feature parity with early NV hardware for a typical desktop Linux user.
ATI's support for video playback acceleration is a very recent (as in, weeks) development in the closed binary blob driver only, and limited to cards with UVD2. It also doesn't work very well for all content, and very few playback apps support the Intel video acceleration API in the first place (ATI and Intel use one API, nv uses their own. The Intel/AMD API looks to be possible as a wrapper around the NV api).
There are multiple other problems with video playback on ATI hardware ranging from TV out support to tearing and flickering in Flash. I won't enumerate them here, but they are a legion strong. The problems encountered by current Linux desktop users are dismissed as not being the target market -- the API is meant for embedded devices.
As far as performance, native Linux 3D games are playable on ATI hardware with most of the 4 series, yes. 5 series is currently having some growing pains. However, most Linux gaming is done with Windows titles and Wine. Wine simply doesn't run for crap on ATI hardware and never has. Yes, the performance is comparatively dismal, but that's the least of it. Visual artifacts (black skin in Oblivion depending on your driver version, e.g.), crashing, lockups. This is dismissed as a Wine problem -- according to ATI the Wine developers appear to be running Nvidia hardware exclusively (no big shocker given the state of ATI drivers on Linux) and as such don't put in ATI-specific hacks.
Most of the Linux compositing desktop effects have started to work recently without problem, but once again with the closed blob driver which doesn't support "legacy" hardware like the 1950XT and older. It's hit and miss, with regressions popping up as often as bugs are fixed. The binary blob may fail to install correctly depending on your distribution. In contrast you can get the nv blob to install on the most customized, homegrown Linux distribution and display these effects flawlessly on the most bleeding edge kernels and Xorg versions using pretty much any product nv made this decade.
So in summary, to a typical desktop user like myself interested in watching video, playing games and using desktop effects eye candy, ATI is not a viable choice today. In six months to a year the picture could be completely different, but I don't have a time machine.