The basic problem is that you're using a POP account, which was designed to be read on one machine. This is why everyone has moved to IMAP, which is designed to be read on multiple devices.
There are two solutions to your problem:
1. You can configure your POP account to hold on to the email on the server for "x" number of days, which gives you time to download it again on the second device. On the second device, you can instruct it to delete the POP mail from the server once retrieved, so that you're not endlessly re-downloading the same mail. However, this won't solve the problem of syncing the mail, it just lets you access it for a period of time on both machines.
2. Forward your POP account to an iCloud, GMail, Yahoo, AOL or other IMAP provider. Problem solved.