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 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.

I don't know for sure, but it would appear he means that it's easier to construct a cubic chain on a product $X \times Y$ 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 and there you really want to map simplices because you'll want to draw a picture of simplex $[a_0, a_1, \cdots, a_n]$ being related to a composition of $n$ categorical arrows, the arrow from $a_0$ to $a_1$ and so on.