I had this problem too, and had about given up on finding the answer. These steps worked for me, and I hope they help you too.
Click Cancel when the wrong password comes up in the popup login window.
Go to Store menu and select View My Account.
Type in the correct password.
Scroll to the bottom of Account Information page and select Done.
Close iTunes.
Re-open.
Or maybe, before closing iTunes, you have to go to Store menu , select Sign Out, and then close iTunes.
When I followed these steps, and then re-opened iTunes, the Store opened without asking me to sign in.
I then went to Store > Sign In and when the popup login window came up, I needed to fill in my username and password. Now, unless I Sign Out, I don't have that login information pop up at all.
I hope this helps. I'm not sure if all of these steps were actually necessarily. You might not need to close and re-open iTunes, but I thought I would just in case.