Is it possible that the computer that works is deleting the email when it checks it? This is the default behavior for POP email clients.
Soo... if you log in with the computer that works, and it downloads the mail, then it will not be there when you check it with the other computer. That's not an error. It's how things work.
If you want the mail to be available on both computers using POP, you have to dig into the server settings on at least one of the computers and find a box labeled "leave a copy of mail on the server" for X number of days, or something along those lines. Usually, most folks do this on the computer that they don't consider their "main" computer.
In other words...
You may have both a computer and a tablet device. You consider the computer to be your "main" email client, meaning it's the one at which you want to make decisions about what mail to keep, what to delete, and so forth. On that computer, you would leave the box un-checked, so when you download the mail, it no longer exists on the server.
The tablet device, on the other hand, you use while working and traveling. You want to be able to see all of the mail, but a lot of it you don't want to deal with until you get home to your comfy computer. So on the tablet, you would check the box to leave a copy of the mail on the server, so it's still there when you get home.
In general, phones used to send and receive mail are also set up to leave the mail on the server.
The other solution, of course, would be to use IMAP, if the provider allows for it. But IMAP's ethos introduces its own set of complexities and dilemmas regarding bandwidth, storage, maintenance, spam-filtering, and other issues; so many providers require (or at least prefer) that you use POP.
-Rich