As is well known, the category of simplicial sets is but one possible 'incarnation' of a (combinatorial) model category among others (CW complexes, other 'shapes', etc.). In contemporary work on homotopy theory and higher topos theory simplicial sets play a key role. We work with categories enriched over simplicial sets, simplicial sets playing the traditional place of the category of sets. The homotopy categories associated to simplicial sets then pass over to simplicial set enriched functors and we can extend analogically many results of the classical theory of Grothendieck topoi, seen as special localizations of presheaf categories.
I read somewhere that Grothendieck was not overly fond of simplicial sets and that his particular vision of 'homotopical algebra' (in Pursuing Stacks) presented an alternative approach. Yet glancing at descriptions of a 'dérivateur' I still find that simplicial sets play a key role.
One argument in favour of simplicial sets is that they occur very naturally as nerves of a category. I don't think you could say the same thing for cubical sets. Also it makes more sense to work with simplicial sets than with their geometric realizations.
Any thoughts on this ?