Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

I think there are many times that simplicial sets are preferable (e.g for classifying spaces the simplicial construction is often advantageous), but to answer the stated question:

• CW complexes connect more immediately to manifold theory (Morse functions give CW structures; a finite CW complex is homotopy equivalent to a manifold by embedding it in some Euclidean space and "fattening it up").
• CW structures can be simpler and more explicit in "small" cases. For example, I do not know an explicit simplicial set whose realization is $CP^2$ (though perhaps I could work one out using a simplicial model for the Hopf map.)
• CW complexes can be analyzed using manifold theory. For example, maps from manifolds to $n$-dimensional CW complexes such as attaching maps can be understood in part by taking a "smooth" approximation and looking at preimages of points in each cell (Goodwillie uses this kind of technique to generalize the Blakers-Massey theorem).

But why should one have to choose "once and for all" between building things from sets vs. from vector spaces, anyways?

My gut reaction is always to work with CW complexes because, being a topologist, I like to work with spaces. Simplicial sets, as nice as they may be, are definitely not spaces.

Both languages are very important. Working with cell complexes you can use geometry, approximations to proof things looking irrational on pure simplicial level. On simlicial language universal constructions looks much better.