When I had nvidia cards I just installed the new driver on top of the old one. Now with an ATI card I am instructed to uninstall the old driver first. Why is this? What would happen if I just install the new driver without uninstalling the old one first?
When installing a new graphics driver without first installing the old there's a risk some of the files from the old driver will remain because they are in use while installing the new driver.
If you look at nVidia's downloads site you will see a link to how to install the graphics driver. This link has actually been there for at least a year and probably longer so it's not only Ati/AMD recommending this. It is also recommended to disable your antivirus program while installing the new driver.
When I had nvidia cards I just installed the new driver on top of the old one.
And same with ATi, they both are similar in that respect, I install overtop of old all the time.... until I have a stability issue, which is still very rare (considering the number of times I install various drivers).
Now with an ATI card I am instructed to uninstall the old driver first. Why is this?
Because that's what you should do with both, especially if you need to ask this question. It reduces the number of conflisct/issues with older driver settings and thus reduces the calls to tech support.
What would happen if I just install the new driver without uninstalling the old one first?
Likely not much, but it could result in partial driver install (crash halfway) or could results in conflicting packages, redundant files (especially in the temp, install and recovery segment), and even poor performance. Usually it's OK, but sometimes it can cause problems, so just like so many things that recommend doing more than the bare minimum, it depends on your comfort level as to what you need to do, but if you want to ensure you system runs as AMD/ATi expect it to, then you want to blow out the old drivers before you put on the new. You can either use the included driver removal package, or a good 3rd party app like DriverCleanerPro.