A new note for people copying their Osu! installs from other computers:
Only copy the songs, Skins, and Replays folders to the new location before installing.
If you install over old installation files, you will not be able to connect, however will be able to log in offline. Solve this by deleting the Data folder and all the files in the main folder and running the install again with the repair option. Your profile will have been deleted, but that doesn't matter, as you should remember your password anyways.
Only copy the songs, Skins, and Replays folders to the new location before installing.
If you install over old installation files, you will not be able to connect, however will be able to log in offline. Solve this by deleting the Data folder and all the files in the main folder and running the install again with the repair option. Your profile will have been deleted, but that doesn't matter, as you should remember your password anyways.