It is always good to know both CW complexes and simplicial sets. Let me give a few examples.

1. To define cup products in some multiplicative cohomology theory, one needs the diagonal map $X\to X\times X$. It respects neither a simplicial nor a CW structure in general, so one has to approximate it. For simplicial sets, you get a certain number of natural choices. One of them leads to the well-known cup-product formula in ordinary singular cohomology. On the other hand, for CW complexes, you just know that such an approximation always exists. If you need a concrete one, you have to work.

2. 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.

3. 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 $\dim M$.

There may even be situations where one wants to combine the strength of both approaches in some hybrid object.