Maybe OP uses some weird mail service which handles POP3 vs IMAP in special ways.
I recall hearing of some servers ignoring deletion by POP3, or automatically deleting all mail downloaded by POP3, or whatever. For the record, I don't even know how POP3 is supposed to work on paper.
With sane mail servers, there is absolutely no problem with multiple IMAP clients. You download to local storage whatever you want, nothing gets deleted unless by your request, you can move messages between folders on the server, messages marked as read on one client show up as read on others - it just works. I have used several mail accounts that way, including private corporate servers and free webmails.