A few things to consider:
Web based mail can be very convenient if you travel a lot. But it can have security problems as all web based sites. If you use a password program like Keepass to generate your passwords then it effectively eliminates the convenience (although, you may put the database of passwords on a memory stick but if you lose the memory stick...).
People often use the same password for different sites. If one site is hacked then all the sites you frequent becomes vulnerable.
Incompatibilities may exist between different email server sites. It depends upon what programming language/platform/toolkit and how they programmed it.
Programs like Pegasus, Outlook, Thundermail, etc. are client-side mail programs whereas web mail are server-side. Client-side needs correct configuration information set up before anything happens. I believe you run a higher risk of losing emails with Client-side programs.
In the past if during transmission the connection is interrupted, you lost the email being transmitted. Or if you had to reinstall the operating system and didn't back up emails, they were lost. Web mail, you keep them indefinitely until the ISP decides to wipe them.
Personally, I use several web based email accounts because its convenient. I use them for storage; email important file as an attachment and address it to me. This is a last ditch backup so to speak. I use a large web based site because I know the backup is going to be there awhile. Important means contacts of companies I deal with. Those companies already have the documents so I don't bother keeping personal information. Its for when my computer crashes.
Its also convenient when surfing the web; just open website in a new tab and read emails. With program based emails (aka client-side), I have to open the program and download it before being able to read it.
p.s. Don't use web based email for storage of personal information. In fact, I won't send personal information through email even under SSL (encrypted). I prefer to use a mobile phone. It seems like every programmer is/can hack a computer but not many have the resources to hack a mobile network. And for the personal information I off-load it to memory sticks.