You need an internet connection. These days this isn’t a problem for many users thanks to reliable broadband. If you are in a region where internet access is unreliable, there’s a risk that you might not be able to access your messages (both new ones and old ones.)
Service isn’t guaranteed. Even the best-maintained web mail service has outages now and again, during which you’ll be without e-mail. However, these are really quite rare, which is why when it does happen there is so much media attention.
The company might lose all the messages! This is theoretically possible. However, in practice, the fear of legal action (and the huge budget available) means a reputable web firm will usually be able to protect your data from loss or damage many times more effectively than you can do on your computer.
Most webmail services are partially supported by advertising. This isn’t generally obtrusive, however.
You may find an increase in spam. This is not really a specific result of webmail; rather it’s because there are so many people using the major services, giving spammers a bigger audience.
You may have trouble getting a good e-mail address. The sheer number of users means ‘your’ name may have already gone and you’ll have to find a potentially clumsy variation. Some people may also think a webmail e-mail address comes across as unprofessional. If this is a worry, check whether an e-mail provider lets you set up and account but receive and send messages using a different address (such as from a domain name, for example firstname.lastname@example.org).
As you’ll have noticed, the list of cons is longer than the list of pros. However, most of the disadvantages are fairly minor and most people will find the advantages far outweigh them.