I haven't used a 62 myself so I can't give specific help but hopefully somebody else will. But with the device connected to your computer, double-click to open it and look in the Garmin directory. Do you see a file named gmapsupp.img? How large is it? This should be the map that you sent. If you can't find that file, the map was not transferred for some reason.
Also, you will need to get Basecamp running somehow because Mapsource is not fully compatible with new units like yours. Mapsource will be fine for sending maps, waypoints, tracks and routes to your 62. However it will not correctly receive them from the GPS. The new units store tracks, waypoints and routes in a bunch of separate files and Mapsource doesn't know where to look for them. Garmin has discontinued Mapsource, so it will never have this capability.
I am running Basecamp on Windows Vista, so it definitely can be done.