Either E-mail application works the same way. You have something installed on your computer, and when you go to a site, it looks at your cookies, sees who you are, and (if you have remember password) will log you into their server, where it will tell you if you have mail or not. Typically, when you are reading an E-mail, it will load some files into your temporary folder, so that you can read it faster (usually pictures, sounds, or links). When you leave the E-mail client, then the cookies and temporary files stay on the computer, but your client can only read them.
Where they differ slightly is when you d/l attachments, where the vast majority of the infections take place.
We are all curious like a cat. We wonder, we ask, we learn.
Please post back when a suggestion works, so that others may learn.