Then it's probably an out of synch login-user mapping. Each Windows account, to include ASPNET, has a SID (Security IDentifier) to identify it uniquely. When you go from one server to another, though the account seems to have the same name, it has a different SID. SQL Server does use that SID when you add the login. And when you add the user, the SID is what matches up the user to the login. So your database has the old SID and your SQL Server has the new SID. This is what is known as an orphaned user in SQL Server.

Check to see if your database has a user by the name of OldComputer\ASPNET or just ASPNET. If either of these exist, then all you need to do is create a new user for the NewComputer\ASPNET login and give it the proper permissions.

In case you encounter this in the future with a SQL Server login, you can use sp_change_users_login to fix things up. Here's a video on how to do so:

Dealing with Orpaned Users in SQL Server