If I remember correctly, the proof of homotopy invariance of singular homology (at least the one in Hatcher) involves cutting a (simplex)x(interval) into simplices, which perhaps can be confusing. In cubical singular homology you'd just have to cut a (cube)x(interval) into cubes, which is obvious. This sort of thing maybe also comes up in other basic results that involve homotopies, such as (graded-)commutativity of cup product.

It doesn't really matter, since as you say, you get the same results at the end of the day. But perhaps in some sense simplices are more "basic" than cubes, since you can easily cut cubes (or any other polyhedron) into simplices, but you can't cut simplices into (finitely many) cubes or even "rectangle-ubes".

As for Ilya's answer, I actually don't think that cubes would necessarily complicate higher category theory stuff that much. It might even make certain things easier. Simplices are just a nice formalism with which to describe things like homotopies, and higher homotopies, and so forth. For example, if you have maps f, g, h such that (f compose g) and h are homotopic, you can think of the homotopy as being a triangle filling in the appropriate diagram. However, suppose you just have maps f and g that are homotopic. Then the homotopy is a "triangle" with a "degenerate" edge filling in the appropriate diagram. So simplices are not perfect either, we still need to allow for "degenerate" situations. In that case, we could have just started with cubes, allowing degenerate edges, to begin with...