I've always thought that all white pasta contains the same ingredients... it is only the shapes that are different.

Is that correct?

Not necessarily. There are rice flour pastas, various kinds of wheat based pastas, and soy based pastas, just to name a few.

Of the wheat based pastas, is the only difference in product the shape?

The wheats can be different.

Some people make whole wheat pastas though typically pasta is made with a semolina flour.

Here is a fairly comprehensive description of wheat flours and others flours:
Cook's Thesaurus: Wheat Flours