I have had this problem before. I cant say for sure I know what causes it, but I have seen a few patterns.
When I had some computers on a domain and some on a workgroup, the ones on the workgroup had to click the drive to mount it. People on the domain generally were able to just log in and access.
This always led me to believe that it was an authentication thing. basically the domain controller in a domain authenticates you so the drive has no reason (or authority) to stop you from accessing.
If you think about it, the authentication would normally occur when you OPEN the drive not just mount it. That is generally why it has no problem mounting when you click it.
As a test I would try the following:
On you local machine, get rid of your passwords. Then set the machine to not require any actual login just boot directly into the windows desktop. (directions at end of post).
On the machine hosting the mounted drive, make sure the user name has no password also.
See if this works. If it does, your solution will either be no password on the local machine, or running a domain to administer it.
AUTOMATIC LOGIN:
1. Click on the Start button and select Run...
2. In the Open box, type control userpasswords2
3. Click OK.
4. A window will appear. Near the top, will be a checkbox for "Users must enter a user name and password to use this computer." To enable automatic login, uncheck this box, and select the account you wish to have automatically login.
