Pop3 + imap

Roderunner said:

Unfortunately Gmail now classes POP3 as not secure, and suggests changing to IMAP.

Do you have some references about that?
I don't think that's really the case (under the proper conditions) it doesn't adds any vulnerability just because of it. For one, GMail forces to use SSL to connect to POP3, so everything is encrypted while in transit. You must use a strong password for access to it, but that also in itself doesn't adds an insecure point, as the very same password is also use though the web interface.
Also, look here: brute force - It's really that insecure to have POP3 enabled in my gmail account? - Information Security Stack Exchange

Roderunner said:

I first tried it using POP3 but !!! so I changed to IMAP.
Now, I get all mail appearing on both pc's making it hard to remember what I have replied to. If I delete one message from AOL on my Laptop, it also gets deleted from my AOL account but not from my Desktop pc.

That's exactly the expected behavior, as POP3 and IMAP work substancially different.
With POP3 you download copies of your mail to your computer (optionally deleting from the server) and everything you see is locally stored. Deleting from the server means that no other client can access them, and also that you have access to your already downloaded mails even if you don't have internet access (everything is local).
With IMAP, you work directly on the server, pretty much as if you were using the web interface. You download mails temporarily, view them on your computer and the discard them. Moving, deleting and everything else happens on the server itself, contrary to POP3 that is a local operation. That's why every client has access to the very same mails, just as if you enter though the web site.

The real question would be, why you can't use POP3 on this account?

Nothing there even mentions POP3. Neither gives any reason for such recommendation.

Moreover, GMail forces to use SSL, which makes sure it encrypts everything while in transit. That, coupled with a good password is reasonably secure. Unless you know of a specific vulnerability, I don't see any reason to worry.