Difference between cellular and simplicial homology Can someone tell me if there is any difference between cellular and simplicial homology?
It seems to me that when I calculate for example the homology groups of the torus it makes no conceptual difference whether I use cellular or simplicial homology. By that I mean: If I think of a zero cell as a vertex, a $1$-cell as an edge and a $2$-cell as an area then these are all simplices. As a consequence, if for example we want to argue that $C_2(T) = \mathbb Z$ then it is enough to either note that there is only one $2$-cell or, equivalently, that there is only one $2$-simplex. 
 A: There are cases when simplicial homology can't be applied but cellular homology can, for example in the case of a compact manifold that can't be triangulated (which exist in dimensions greater than or equal to 4).
A: All the cellular and simplicial homology groups of any (triangulable) space are isomorphic. For a proof see e.g. Hatcher.
A: As explained in another answer here, simplicial homology can be regarded as a special case of cellular homology;  but the standard definition of the latter  and establishment of its  properties requires the main facts on singular homology.
For a short discussion of some history of algebraic topology, and some anomalies,  relevant to this question see this  presentation.
The book partially titled Nonabelian Algebraic Topology (EMS Tract Vol 15, 2011) takes a different approach to homology in that,  starting with a filtered space, $X_*$, for example the skeletal filtration of a CW-complex, we define directly and  homotopically,  using fundamental groupoids and relative homotopy groups , a "crossed complex" $\Pi X_*$,   which is a kind of  chain complex with operators, but partially, i.e. in dim $\leqslant 2$, nonabelian. Essential freeness results on this for the CW-filtration case are proved using a Higher Homotopy Seifert-van Kampen Theorem for $\Pi$; this theorem has an intuitive but roundabout  proof, using cubical groupoid methods to give "algebraic inverses to subdivision".
This approach replaces by actual compositions,  i.e. higher dimensional analogues of compositions of paths,  the "formal sums" standard in the usual approach. It  allows for  results which are nonabelian in dimension $\leqslant 2$, and are on modules in  dimensions $> 1$, the latter without using covering spaces. 
Part of the inspiration for this approach comes from J.H.C. Whitehead's 1949 paper "Combinatorial Homotopy II". 
