AquaMail syncs with the server, not with the other mail app directly...
Are the messages sent in the other mail app available on the mail server? You can use web mail to check.
Secondly, is AquaMail set up to sync [ Sent ] with the server? To check, long press the account, choose Options and Folders -> Folders, scroll to the server's Sent folder, make sure it says "Sync with [ Sent ]". If not, tap and change the option.