Real thin clients (as with LTSP) are awesome and pretty much gives you all the benefit of thin clients and fat client combined; it even allows access to local hardware which allows you to run 3D graphics, use local sound, USB disks etc without having to do some weird protocol hacks. These days it's even aware of remote apps so if you choose to open a PDF in your web browser running as a local app you can have it open it on a remote server if you want to.
If your 'thin client' is powerful enough you can also choose to run everything locally, essentially making it a fat client that just uses the network as a filesystem. This isn't particularly useful for systems like laptops, but for libraries, schools, etc that wish to minimise maintenance and support, it's awesome.
Also, "thin client" doesn't imply VDI, and fat and thin client infrastructure aren't mutually exclusive, there are tons of configuration managers out there that allows you to easily keep your fat client and application servers running the way you want to. In most environments it's probably a good idea to have a mixture of both.