Turing complete simply means when a computational engine can mimic the rules of a Turing machine. On Computable Numbers, with an Application to the Entscheidungsproblem by none other than Alan Turing explains the concept of a Turing machine. Once something is Turing complete, you can no longer make simple assumptions about what it will do (or hell, if it will even halt).

TV channels are quite restricted on what they can do (they can put frames of images on your screen and maybe some metadata). So safer assumptions about what happens when you funnel a video stream to a TV can be made. Computer programs are not like TV channels..