Many topologists express a clear preference for working with CW complexes instead of simplicial sets.

One of the reasons is that the cellular chain complex of a CW complex is often easier to work with than a simplicial chain complex. However, simplicial sets have many nice features that spaces do not. The category of simplicial sets has a proper and combinatorial (in the sense of Jeff Smith) model structure and is a presheaf topos, which makes the objects behave very much like sets. Surely these make up for the problems with specifying combinatorial data?

The question: Why do many topologists and homotopy theorists prefer to work with spaces and CW complexes over simplicial sets and Kan complexes? What are some other advantages that CW complexes enjoy over Kan complexes?