I don't know for sure, but it would appear he means that it's easier to construct a cubic chain on a product `X x Y` given cubic chains on `X` and `Y` compared to the simplex chain given two simplex chains.

Since it's a simple exercise to go from simplices to cubes and vice versa, I don't see any advantage to cubes. In fact, I would expect any good book to explain that *either* of approaches could be used.

It would unnecessarily complicate the notation in the modern retellings of higher category theory, though. I learned about it from Lurie, *Higher Topos Theory*, [0608040][1] and there you really want to map simplices because you'll want to draw a picture of simplex `[a_0, a_1, ..., a_n]` being related to a composition of `n` categorical arrows, the arrow from `a_0` to `a_1` and so on.


  [1]: http://arxiv.org/abs/math/0608040