I just moved to a new laptop myself and it's basically like frankz says. I made sure that my username was the same on both systems, I copied over all the audio files from Music to Music, and I copied three specific folders to the new computer:
The Library folder (where the .mbl is)
The AppData folder
The Program Files folder
I did this after installing MB on the new system, so I could be sure I was overwriting the files that were created during installation. When I next started MusicBee up, it was exactly where I'd left it on the old system, right down to the spot that the currently playing track was paused.