Here is another interesting class to throw in the mix in between CW-complexes and  general spaces. They are defined exactly like CW-complexes, by inductively attaching cells, except that you are allowed to attach cells in any order. I think these have the homotopy type of CW-complexes, but are nicer point-set wise then the general class of things with these homotopy types. (See Tyler's answer).

They also show up in nature! If you have a manifold and a Morse function, then the handle body structure doesn't usually give you a CW-complex structure unless the critical values are ordered by index (i.e all index k critical points have the same critical value, say k). Otherwise you end up attaching cells in unusual orders.  

However, a CW-complex has a natural *filtration*. This allows you to induct on the dimension of cells. It also allows you to construct certain easy spectral seqeuences. For example, the cellular homology complex comes from a sort of trivial application of this filtration. More generally the construction of the Atiyah–Hirzebruch spectral sequence uses this filtration.  So it is nice to know something is a CW-complex, or at least is homotopy equivalent to it. 

So to summarise:

 1. CW complexes have nice point-set and homotopical properties.
 2. CW complexes have nice computational properties (for example a useful filtration).
 3. Knowing that $X$ is homotopy equivalent to a CW complex allows you to transfer computational (homotopy invariant) results about CW-complexes to $X$. For example: the Atiyah–Hirzebruch spectral sequence.

There are many other classes of spaces satisfying the first condition, but fewer also satisfying the second. But notice that once you *have* the class of CW complexes you can do all sorts of things for spaces that are only homotopy equivalent to CW complexes without them actually being CW-complexes. This is a subtle point. Once you have CW-complexes at your disposal you can prove results about things which are not themselves CW complexes, which would otherwise be difficult to prove.