I personally always get widescreen. They used to call that letterbox, but the term had died down a bit.
The difference is in the shape of your TV compared to the shape of a movie screen. Movie screens are much wider then your TV, which is where they get the different aspect ratios. Different directors will select different aspect ratios either because of artistic preference or camera limitations.
Widescreen preserves the aspect ratio that the director uses. On full screen, the image is cut off slightly on the sides, and extended very slight length-wise, in order to fit a typical TV. Most will not notice that much of a difference, but it does change what the director intended. In certain movies, the director will want two characters to look like they are not very close to each other, however in full screen because of the streching of the film they do appear a little closer then inteded (Raging Bull is the best example).
Another example is in the original Star Wars after Luke meets Obi-Wan - they point out one of the sandpeople, and there is a cut to the desert. In widescreen, you see the guy on the edge of the screen. In full screen, the guy has been cut out.
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