Aqua Mail does not delete messages from mail servers just when it checks mail.
Only when you tell it to (by moving messages to Deleted which will be reflected this way in web mail and mail server if the account is set up as IMAP... If it's set up as POP3, messages will be deleted when you empty Deleted).
And then, some mail servers will let only one app, only once, see any particular message -- if using POP3. One such mail server is Gmail.
So if you're trying to use Gmail over POP3, please don't, and switch all your mail apps to using IMAP.