ISPs normally restrict the use of their email services to their subscribers. If you switch to a different ISP you lose access to their service. (Their POP3 and SMTP servers, which handle incoming and outgoing mail respectively, check to see whether you're a subscriber when you try to access them, whereas the web server you seem to be using to view old mail doesn't do so).
Some ISPs cut off both incoming and outgoing services at the moment you leave them. Others allow a 'grace period' (which might be anywhere from a few days to several months), with a few allowing incoming (but not outgoing) mail indefinitely.
Forget about AOL (which was never a good idea in the first place!). Get yourself a decent free email account with GMX (which will remain active however many times you change your ISP):
http://www.gmx.co.uk/
You can access your GMX account using webmail (i.e. by going to their website) or, if you prefer, by using a dedicated email client (such as Outlook Express or Windows Live Mail). Probably the best non-Microsoft client (and certainly the best-known) is Thunderbird:
https://www.mozilla.org/EN/thunderbird/
Chris