Ubuntu – Unity does not start after booting

I'm using a relatively new installed Ubuntu 13.04 with all updates. My video driver is installed from the official AMD website and worked before without trouble.

After the latest update, Unity did not start anymore (I think). The launcher and the top bar did not appear, so the only thing I could see was the background and the desktop icons (so nautilus was useable). But there was no Window frame around the windows, so I was not able to see the close-button and so on. I thought this is probably caused through the video driver (installed using amd-catalyst-13.4-linux-x86.x86_64.run installer file). So I ran it again.

Now, the frames around the windows are back. But the rest stayed as before.

Is it possible to get everything working again without installing everything new? I would also use the OpenSource or the proprietary video driver from software-properties-gtk, but it does not allow me to "downgrade" or choose the open source driver.
If you need any additional information just tell me.

Thank you.

PS: I can imagine this was already answered, but I could not find any working answer and it would be good if I could use my computer normally soon.

Update:
I changed to the open source driver using the instructions from an answer below, but functionally nothing changed. Unity still does not show up.