The best way to explain the difference betweeen web mail and email software is to use a post office analogy. Web mail is a mail box address located on a remote server – Hotmail or Yahoo Mail, for example. To read your mail and send messages you have to go to the web mail server and access your mail box while you are there. This would be like going to the post office to get your mail from a P.O. box.
Email software is more like the letter carrier who delivers your mail to your door. You tell the email program where your mail box is located (in the account settings of the program) and it goes to your mail box and fetches the mail back to your computer where you can save it for future reference.
Web mail is great when you are traveling because you can access your email from any internet-connected computer anywhere in the world. For collectors, like me, having email come to your computer where you can manage your email without any space restrictions is the way to go.