The xorg open source driver is far more stable than the proprietary one, so if you don't need any of the features of the proprietary driver, stay with the open source driver.
Fglrx and fglrx-updates are the same when a new version of Ubuntu is released. Fglrx-updates later is updated when newer drivers are released by AMD, while fglrx is in general not updated until you upgrade to a new release of Ubuntu. This means, that fglrx-updates will probably have better performance, less bugs and more features than fglrx, yet it might in principle happen, that an update is released for it, which contains new bugs (which might even prevent your system from starting a GUI).
Now to the advantages of the proprietary driver:
- It has far superior 3D performance, what is also relevant for desktop performance when using a compositing window manager (Unity uses Compiz, which greatly relies on 3D acceleration)
- With it the graphics card uses less power (mainly relevant on laptops)
- It comes with OpenCL support, meaning that some programs can benefit from the graphics chip (for instance imagemagick, although I think this feature is disabled on Ubuntu)
- You can use VAAPI to get hardware accelerated video decoding (for instance in VLC media player), but this usually is only required on very slow CPUs, like AMD C-series or Intel Atom.
- On some cards, the open source driver does not support audio output over HDMI, in this case, use the proprietary driver.
Now the disadvantages of the proprietary driver
- Did I mention that it's less stable than the open source driver?
- On some systems tearing artifacts are visible when playing video files (there is a setting in the Catalyst Control Center to prevent this, yet it doesn't work on some systems)
- For some settings one has to use Catalyst Control Center instead of Ubuntu System Settings, for instance if you want to extend the desktop to a second monitor for the first time.
Since they are mentioned in one of the comments, just a word regarding the experimental drivers: I don't recommend to use them, they are at the moment (at least on 12.04) incompatible with some other packages (most notably: hardware video decoding and opencl don't work without ugly hacks). If you really need the latest drivers from AMD, get them from the AMD website and use them to build distribution specific packages.
Long story short: If everything works fine for you with the open source drivers, use these. If you want or need one of the features mentioned above, and you are ok with the very small risk that an update breaks something, use fglrx-updates, otherwise fglrx. Don't use the experimental drivers if you are not sure what you are doing.