The major one is you lose all your emails when your webmail provider shuts down.
A lot of people switched to free email services so that they wouldn't lose their email address when they switch providers (may that be a consumer ISP or institutional provider). In this respect, I suspect most people benefited. The popular services have been available for approximately 10 to 20 years. How many people have had the same ISP for that time?
(Yes, I know there are other ways of handling this. None of those ways are particularly cheap or accessible. On top of that, at best, they just insulate you from a provider shutting down. That is to say, you still have to scramble to find a new provider when one shuts down even though you may be able to keep your address.)