Why not filming computer or TV screens?
There are several factors that comes to play with recording computer screens.


As filzilla already mentions, sync is a major factor. In the old days CRT monitors was very challenging to film even with sync-locking. Depending on if you filmed on silver or on video you did get different results. With the exception of LED screens, LCDs also offer sync problems.
The other factor is light conditions. For movie scene you usually have a large amount of light which a computer screen is unable to produce - so the screen turns very dark. You also get problems with color-correction/white-balance as the monitor typically have a different color temperature than the lights used to lit the scene.
The third factor is of more recent nature and has to do with the nature of LCD screen and it's polarization, referring to the limited angle they are visible within. You will get "gradient" effects and on more poor screens also have limited visibility from the sides, top and bottom.
Moiré can be a problem. For video this isn't such a big problem because the light coming from the screen will blend the pixels. Unless you have a high resolution camera filming in low(er) light conditions, or are filming close-up (but not so close that you can actually see the screen pixels), the camera won't be able to separate the individual pixels. For movie cameras this is more present though due the higher resolutions and the requirement to light conditions.
Then there is content synchronization with the actor's interactions or the scene in general, the potential of typos, wrong clicks, having to reset for each take and so forth.


It's more easy to put in a video in post which you can color correct and adjust, synchronize and so forth to the movie.

The process isn't very complicated. The computer screen is often recorded with a green or blue backdrop to eliminate the need for mask/matte rotoscoping. You just track the corners of the inner part of the screen and drop in a keyed replacement for the screen.

There are plugins too that do a pretty decent job automating this process (or instance the Mocha AE and its planar tracking).