Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets? Many topologists express a clear preference for working with CW complexes instead of simplicial sets.  
One of the reasons is that the cellular chain complex of a CW complex is often easier to work with than a simplicial chain complex.  However, simplicial sets have many nice features that spaces do not.  The category of simplicial sets has a proper and combinatorial (in the sense of Jeff Smith) model structure and is a presheaf topos, which makes the objects behave very much like sets.  Surely these make up for the problems with specifying combinatorial data?
The question: Why do many topologists and homotopy theorists prefer to work with spaces and CW complexes over simplicial sets and Kan complexes?  What are some other advantages that CW complexes enjoy over Kan complexes?
 A: Both languages are very important. Working with cell complexes you can use geometry, approximations to proof things looking irrational on pure simplicial level. On simlicial language universal constructions looks much better. 
A: I feel topologists should be eclectic and able to move across a number of models, to see which better aids understanding. Here is a part quotation from Einstein, which I feel also reflects the concerns of the questioner:
"What is essential and what is based only on the accidents of development?... Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. ....It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little..."
Simplicial sets have a very well developed theory, are "convenient" in many ways, but they have limitations. The results of the book Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical omega-groupoids (NAT) (pdf available) would not even have been conjectured simplicially, but the notion of multiple compositions of cubes led to their conjecture and proof. A search on "cubical" in mathoverflow gives more relevant information. To go back to the question, the above book gives a new outlook on structures related to CW-filtrations, and the border between homotopy and homology, using cubical sets (with connections). 
See also this brief 2015  presentation on "A  philosophy of modelling and computing homotopy types": aveiro.  
January 5, 2016 
Since products are referred to in other answers, I mention that the isomorphism $$C_*(X_*) \otimes C_*(Y_*) \cong  C_*(X_* \otimes Y_*)$$   in the cellular case is extended in this case   to an isomorphism $$\Pi(X_*) \otimes \Pi(Y_*) \cong \Pi(X_* \otimes Y_*) $$ in the   above NAT book. Here $\Pi$ is a homotopically defined functor on filtered spaces with values in crossed complexes: this functor contains information on  relative  homotopy groups $$\pi_n(X_n, X_{n-1},x), x \in X_0, n \geqslant 2,$$ and  on the operation of fundamental group(oid)s on these. Proofs use cubical methods, relying on the isomorphism $I^m_* \otimes I^n_8 \cong I^{m+n}_*$.  
A: I think there are many times that simplicial sets are preferable (e.g for classifying spaces the simplicial construction is often advantageous), but to answer the stated question:


*

*CW complexes connect more immediately to manifold theory (Morse functions give CW structures; a finite CW complex is homotopy equivalent to a manifold by embedding it in some Euclidean space and "fattening it up").

*CW structures can be simpler and more explicit in "small" cases.  For example, I do not know an explicit simplicial set whose realization is $CP^2$ (though perhaps I could work one out using a simplicial model for the Hopf map.)

*CW complexes can be analyzed using manifold theory.  For example, maps from manifolds to $n$-dimensional CW complexes such as attaching maps can be understood in part by taking a "smooth" approximation and looking at preimages of points in each cell (Goodwillie uses this kind of technique to generalize the Blakers-Massey theorem).


But why should one have to choose "once and for all" between building things from sets vs. from vector spaces, anyways?
A: It is always good to know both CW complexes and simplicial sets. Let me give a few examples.


*

*To define cup products in some multiplicative cohomology theory, one needs the diagonal map $X\to X\times X$. It does not respect a CW structure in general, so one has to approximate it. One knows that such an approximation always exists. If you need a concrete one, you have to work. For simplicial sets on the other hand, the diagonal map is simplicial. But then it is harder to relate $h^\bullet(X\times X)$ with $h^\bullet(X)\otimes h^\bullet(X)$.
One of the possible solutions for ordinary cohomology leads to the well-known cup-product formula in singular cohomology. It is interesting to notice that this cup-product formula looks as if it came from an approximation of the diagonal in the CW-product of the geometric realisations.

*Every topological space has an approximation by a weakly homotopy equivalent CW complex. If you are lucky, you find one with very few cells, for example a single point suffices for the polish circle. But there always is a natural choice, the realisation of the singular complex $\left|S_\bullet(-)\right|$, which is awfully large in most cases. However, in the case of the polish circle, you may argue that a weakly homotopy equivalent approximation loses too much information, and prefer to use some entirely different theory.

*Starting from a smooth manifold $M$, a Morse function together with a gradient-like vector field gives a CW complex, which is again far from natural. But you immediately recover the dimension of $M$ from the cell structure (which you can then use to prove a cup-length estimate, for example). The singular simplicial complex has nondegenerate simplices in all dimensions, so you really have to work until you recover $\dim M$.

*Yet another point. For any topological group $G$, Milnor's join construction gives a model for the classifying space $BG$ that is a simplicial space. It is "made" to classify $G$-bundles gives by $G$-cocycles. On the other hand, if $G$ is a classical Lie group, you can approximate $BG$ through Grassmannians. These classify vector bundles that are given as subbundles of trivial bundles (which is how you usually view vector bundles in noncommutative geometry). The construction is again less universal, but it connects better with some analytic methods. And one has Schubert cells to work with.
There may even be situations where one wants to combine the strength of both approaches in some hybrid object.
A: My gut reaction is always to work with CW complexes because, being a topologist, I like to work with spaces.  Simplicial sets, as nice as they may be, are definitely not spaces.
