Pros
Once you get your own clients, you start to become your own boss. You decide how to best manage a site, how to improve rankings, and how to get better ROI. You decide when to take lunch or how much overtime you want. If you need help, there are plenty of people to ask. You can have a lot of control over your day. With all the different sites you work on, there are lots of really cool things to learn about, which in turn help your clients.
Cons
While being independent is great, there are times things get lost in communication. A designer will imagine one thing for a site while the programmer planned another leaving you out of the loop, a series of emails won't have all the right people CC'd, etc. Clients can be the biggest offenders here, waiting until the absolute last second to respond.