There are probably several answers to this. Here's my take.
Two things make the classification of von Neumann algebras interesting and useful, in my view:
1) After you define the types, the abundance of projections allows you to show that any von Neumann algebra is a direct sum of subalgebras of some of the types.
2) There are many cases where the type information on its own tells you a big deal about the algebra: I'm thinking of results like:
- Type I factors can be completely classified;
- Type II$_1$ factors always carry a faithful normal tracial state;
- Type II$_\infty$ factors are always a tensor product of a II$1$ and a I$\infty$;
- Type III factors are a crossed product of a II$_\infty$.
- AFD factors can be completely characterized for all types.
For C $\!\!^*$-algebras, one can try to play the same game (for example, "simple" could play the role of "factor", Type I C $\!\!^*$-algebras, purely infinite versus finite, AFD, etc.), but one is immediately hampered by the (eventual) lack of projections, that forbids to always have a C $\!\!^*$-algebra as a direct sum of simpler ones.
As a final word, "classification" is also used as in Elliott's Classification Program. In this setting, it is not clear at all that von Neumann algebras are on better footing that C$^*$-algebras. Of course type I von Neumann algebras can be completely classified, and rather easily; but, for example, a complete classification of all II$_1$ factors is considered completely hopeless by all experts.