Our new book
uses mainly cubical, rather than simplicial, sets. The reasons are explained in the Introduction: in strict cubical higher categories we can easily express
algebraic inverse to subdivision,
a simple intuition which I have found difficult to express in simplicial terms. Thus cubes are useful for local-to-global problems. This intuition is crucial for our Higher Homotopy Seifert-van Kampen Theorem, which enables new calculations of some homotopy types, and suggests a new foundation for algebraic topology at the border between homotopy and homology.
Also cubes have a nice tensor product and this is crucial in the book for obtaining some homotopy classification results.
I have found that with cubes I have been able to conjecture and in the end prove theorems which have enabled new nonabelian calculations in homotopy theory, e.g. of second relative homotopy groups. So I have been happy to use cubes until someone comes up with something better. ($n$-simplicial methods, in conjunction with cubical ideas, turned out, however, to be necessary for proofs in the work with J.-L. Loday.)
See also some beamer presentations available on my preprint page.
Here is a further emphasis on the above point on algebraic structures: consider the following diagram:
composition http://pages.bangor.ac.uk/%7Emas010/composition2.jpg
From left to right pictures subdivision; from right to left pictures composition. The composition idea is well formulated in terms of double categories, and that idea is easily generalised to $n$-fold categories, and is expressed well in a cubical context. In that context one can conjecture, and eventually prove, higher dimensional Seifert-van Kampen Theorems, which allow new calculations in algebraic topology. Such multiple compositions are difficult to handle in globular or simplicial terms.
The further advantage of cubes, as mentioned in above answers, is that the formula $$I^m \times I^n \cong I^{m+n}$$ makes cubes very helpful in considering monoidal and monoidal closed structures. Most of the major results of the EMS book required cubical methods for their conjecture and proof.
Sept 5, 2015: The paper by Vezzani arxiv::1405.4548 shows a use of cubical, rather than simplicial, methods, in motivic theory; while the paper by I. Patchkoria, HHA arXiv:1011.4870, Homology Homotopy Appl. Volume 14, Number 1 (2012), 133-158, gives a "Comparison of Cubical and Simplicial Derived Functors".
In all these cases the use of connections in cubical methods is crucial. There is more discussion on this mathoverflow. For us connections arose in order to define commutative cubes in higher cubical categories: compare this paper.
See also this 2014 presentation The intuition for cubical methods in algebraic topology