Fat clients are generaly easier to build as you can go directly to the database, and you get to do everything on the users desktop. Fat clients are great if you have a lot of bandwidth between your servers and your users.
Thin clients are good for customer facing applications as they are ususally web based. They are also good when the network link between the users and the servers is slower.
Unless you are talking about thin clients as workstations. In that case…
Fat clients are nice because the user has the entire OS to work with, and they can do anything that they need to. This does create a support problem as users can do anything they’d like (that they are alowed to do). When using fat clients you do need some way to update users applications such as SMS.
Thin clients are great for support becuase the users just RDPs into a terminal server. If the application ever needs upgrading you simply just upgrade a single machine any everyone has the new version.