Why it works from one computer to another, but not the opposite way: you granted permissions for the "everybody" group on the computer that can be accessed easily. And possibly, you didn't deactivate the "guest" account. My guess the computer it's asking you the guest password. [LE] To be read as: My guess the computer it's asking you the guest password. -- it's about the computer that it is not "easy" to browse[/LE]
If you want to grant yourself access to folders on the other computer you have to play a little with user rights/permissions on that computer. If you want to share just to you username, let's say Iam, on the computer TheComputer, and you want to access from another computer, OtherComputer you have to:
1) grant permissions to user OtherComputer\Iam (in case that the account Iam exists only on OtherComputer) to the folder needed, and the share needed. Be careful, you have to adjust both security settings (folder and share). But, if I am not mistaken, the security options for share are "everyone -> full access", so you'll have to work only with folder security settings.
2) enter on TheComputer (Start->Run->\\TheComputer ip address), it will ask you for a user and a password. If Iam is defined on TheComputer, you will enter as username Iam, with the known password. After that you'll see a list of shared folders.
3) if the Iam account exists on TheComputer and OtherComputer, the same rules as above, except that you can touch your share directly, with Start->Run->\\TheComputer ip address\your_needed_share.
Pay attention: you have to play with folder security options, eventually with share security options. It's not difficult, in a few (20-30 minutes) minutes you'll be able to understand and apply the rules concerning the security options for local folders and shares.
Tell us if you succeeded.
P.S. These explanations only for Windows 2000/ XP/ 2003