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 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.

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.

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 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.

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 $\dim M$.

4. 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.

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.

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 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.

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.