I think this question is an interesting one, and the two approaches should be compared. I have more knowledge of Whitehead's programme, and have been involved with others in the development of aspects of that programme. A report on the background of part that work is in the paper Modelling and Computing Homotopy Types: I, to appear in 2017 in a special issue of Indagationes Mathematicae in honor of L.E.J. Brouwer. A notable feature of this work is that it deals not with "bare" spaces but with "Topological Data", and this somewhat reflects Grothendieck's view expressed in "Esquisse d'un Programme" Section 5. Also among the aims of Whitehead's work was to introduce invariants which allowed specific calculation and even if possible enumeration.
I should explain that having from 1965 to 1974 tried to define for a space $X$ a strict homotopy double groupoid which could satisfy a van Kampen type Theorem, and so enable specific calculations, it was a considerable relief to find with Philip Higgins that this could be done in a natural and intuitive way for a pair $(X,A,c)$ of pointed spaces by mapping the square $I^2$ into $X$ so that the edges of $I^2$ mapped to $A$ and the vertices mapped to $c$. This gave a vast generalisation of a tricky theorem on free crossed modules, proved in Section 16 of the 1949 paper "Combinatorial Homotopy II" (CHII).
Further this definition of homotopy double groupoid generalised, with considerable more work, to all dimensions using filtered spaces, so continuing Whitehead's programme in CHII.
The algebraic data used here of strict cubical $\omega$-groupoids, and the equivalent crossed complexes, do not model all homotopy types, and indeed contain in essence only "linear" information, i.e. no Whitehead products, for example. However these models do contain more information than chain complexes with a group of operators, conforming with an observation of Whitehead in CHII.
However a meeting with J.-L. Loday in 1981 in Strasbourg started our link with his work on what he called $n$-cat-groups, and we later agreed to call cat$^n$-groups, and which are $n$-fold groupoids in which one direction is a group. Loday had proved in 1982 modelled pointed homotopy $(n+1)$-types. We conjectured then and eventually proved a van Kampen type theorem in the context of $n$-cubes of spaces; this work was eventually accepted for the journal Topology, and a companion paper was accepted for Proc. London Math Soc, both appearing in 1987. The latter paper proved an $n$-adic Hurewicz Theorem, the triadic version of which was a conjecture of Loday in 1981.
Actually Grothendieck objected to the pointed condition, and did not recognise what had been achieved - he always wanted the most general conditions! However one aspect of the work with Loday, a nonabelian tensor product of groups which act on each other, has been well taken up by group theorists, and a current bibliography has 158 items dating from 1952.
H.-J. Baues in several books has continued aspects of Whitehead's programme, but he does not apply the Brown-Loday work, and his models do not satisfy the Criteria set out in Section 1 of the cited paper "Modelling and Computing Homotopy Types:I". However paper II of that has been delayed for various health reasons.
This paper, in a volume in memory of J.F.Adams, refers to work of G.J.Ellis and R. Steiner which applies the Brown-Loday work to solve an old problem in homotopy theory, on which Whitehead had written, to determine the value of the critical (i.e. first non vanishing) group of an $(n+1)$-ad.
Looking again at Esquisses d'un Progamme, it seems that programme has currently little relation to Whitehead's; but a 1983 letter from Grothendieck to the writer, reprinted as Problem 16.1.29 of Nonabelian Algebraic Topology, may suggest that despite his interest in the 2-d van Kampen theorem, the lack of progress with the issues he there raises indicates a limitation of this writer.
Grothendieck was very interested at one point in the idea I once conveyed to him that $n$-fold groupoids model homotopy $n$-types. But this is not true as stated even for $n=2$, since $2$-fold groupoids can be much more complicated than crossed modules over groupoids, which are a good model of homotopy $2$-types, and are equivalent to a special kind of double groupoid.
Added 18/04/2017: A final remark is that in my work with Higgins the cubical aspect is essential, and analogous results have not been obtained by simplicial methods. However the work with Loday does use also advanced simplicial methods for the proof of the main result. In both cases, the cubical methods are used to express multiple compositions, which are tricky simplicially.