A hefty price premium (when directly compared to CRT monitors) and certain limitations that can affect fast frame rates in the current crop of todays games are the most obvious downsides.
Some people seem to have issues with font clarity (I don't on any of my TFT screens) while others report pixellation when working close in on high resolution images in Photoshop or similar image editing software.
Aside from that, in very general terms, there's not really too much against them, but there is a very great deal for them.
Keep in mind that there are some quite marked variations in performance within a range of products from different manufacturers. Some offer better image quality than others, so read a comparative review of several products before parting with any cash.