In terms of Pros and Cons, You'd probably be able to find a desktop PC cheaper then an equivalent spec'd laptop. The only other difference would be the portability of a laptop over the desktop.
In terms of connecting your PC to a TV, as long as they have the same inputs and outputs its entirely possible. Ideally you'd have a TV and PC with HDMI ports as you'd only need one cable for audio and visual. You can also do the same with a laptop and TV with the added possibility of carrying it away if you need to.