Try logging out and back in (this is not the same as restarting STEAM)
To do this:
1. Make sure you remember you're password. Username was saved for me.
2. Open STEAM
3. Select File
4. Select change user
5. A message will come up saying this will log you out of STEAM.
6. Click OK
7. Enter correct details
8. Log in.
Now after this is complete try getting into a server.