What does actually being a CW-complex provide in algebraic topology? From time to time, I pretend to be an algebraic topologist.  But I'm not really hard-core and some of the deeper mysteries of the subject are still ... mysterious.  One that came up recently is the exact role of CW-complexes.  I'm very happy with the mantra "CW-complexes Good, really horrible pathological spaces Bad." but there's a range in the middle there where I'm not sure if the classification is "Good" or just "Pretty Good".  These are the spaces with the homotopy type of a CW-complex.
In the algebraic topology that I tend to do then I treat CW-complexes in the same way that I treat Riemannian metrics when doing differential topology.  I know that there's always a CW-complex close to hand if I really need it, but what I'm actually interested in doesn't seem to depend on the space actually being a CW-complex.  But, as I said, I'm only a part-time algebraic topologist and so there may be whole swathes of this subject that I'm completely unaware of where actually having a CW-complex is of extreme importance.
Thus, my question:
In algebraic topology, if I have a space that actually is a CW-complex, what can I do with it that I couldn't do with a space that merely had the homotopy type of a CW-complex?
 A: The words "actually is a CW-complex" suggest to me that the CW-structure is known, whereas simply having the homotopy type of a CW-complex suggests the CW-structure is unknown, or at least not uniquely determined. So one answer to your question could be, "compute invariants that are defined using the CW-structure".
I have in mind cellular homology and cohomology, of course. But there are other examples from homotopy theory, such as various types of Hopf invariant, boundary maps in cofibre sequences, the Leray-Serre and Atiyah-Hirzebruch spectral sequences, $\ldots$
$\ldots$ and Whitehead torsion (see John Klein's answer).
A: There is a fundamental reason why CW-complexes $X$  say  are useful, namely for the purpose of constructing continuous functions $f: X \to Y$ by induction on the skeletal filtration $X^n, n \geqslant 0$,  and for going from $X^{n-1} $ to $X^{n}$ knowing that you have to construct the extension only on each $n$-cell $e^n$. The topology on $X$ is arranged precisely for this purpose. So a complicated space is put together from simple ones. 
It is this possibility of constructing continuous functions, and also homotopies, which allows for the proof of the Theorem of Whitehead mentioned above. 
It interesting to trace the development of this concept from his earlier papers for example
Whitehead, J H.C. On incidence matrices, nuclei and homotopy types. Ann. of Math. (2) 42 (1941) 1197--1239, 
which introduced the term "membrane complexes, which are so to speak more elastic than simplicial complexes". The idea was to avoid the subdivision into simplices by amalgamating them.  This paper, and others, were rewritten after the war, in terms of the CW-complexes we know and love. 
A crucial point was to be able to construct not only continuous functions but also continuous homotopies of these. For these a result on the product of CW-complexes was needed,  which I was told took him a year to prove. He also early on formulated basic results on adjunction spaces, which are now clear in terms of pushouts.  
I feel the emphasis on filtrations is crucial, and allows for other methods in algebraic topology. 
22 Nov 2013: Grothendieck in his "Esquisse d'un programme" Section 5 (English version available here, see p.258) discussed what he feels is the inadequate nature of topological spaces for certain geometric considerations. 
When foundations are under consideration, one should take a "no holds barred" attitude. This agrees with the advice of Einstein: 
"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. They then become labelled as "conceptual necessities", "a priori situations", etc. The road of scientific progress is frequently blocked for long periods by such errors. 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..." 
A: Those of us who do algebraic topology too much should remember occasionally that topological spaces are, in general, terrible to work with.
CW-complexes have a lot of properties that make them nice to work with in homotopy theory, such as being amenable to study by homotopy groups and such as being able to define maps out inductively.  These properties are still possessed by objects with the homotopy type of a CW-complex.
However, CW-complexes are also nice on the point-set level.  They're compactly generated, locally contractible, and every compact subset is contained in a finite CW-subcomplex - and finite CW-complexes have almost every space-level regularity property that one can name.  If one has the goal of doing homotopy theory, the fact that we have this large class of nice spaces means that (to a certain extent) we can set many point-set considerations aside.  And sometimes these point-set considerations can be irritating (such as the smash product being nonassociative).
Objects just in the homotopy type can be terrible.  Take the cone on your favorite pathological space and you find something with the homotopy type of a nice CW-complex but terrible point-set behavior.
A: As a homotopy theorist, the best reason for me is Whitehead's Theorem. This says that if $X$ and $Y$ are connected CW complexes and $f:X\rightarrow Y$ is a weak homotopy equivalence (i.e. induces an isomorphism on $\pi_n$ for all $n$) then it is a homotopy equivalence. According to wikipedia, this was the original justification for CW-complexes when Whitehead introduced them. I'd say this is the best answer I can give for the question of "what can you actually do with CW complex that I can't do with a general topological space."

EDIT: Here are other reasons CW Complexes are cool, but these don't seem to directly answer the question in the way that the above does.
Another reason to consider CW complexes is that they are much easier to work with than general topological spaces because of the inductive definition using cells. In particular, a CW complex is a colimit of its $n$-skeleta $X_n$. This makes it much easier to compute things for CW complexes, e.g. cellular (co)homology, homotopy. Moore spaces and Eilenberg-Maclane spaces are CW-complexes and the constructions aren't too hard. Also, you can construct a space $X$ as an inverse limit of Eilenberg-Maclane spaces via the Postnikov Tower, so again CW complexes give you a way to "get your hands on" a general space $X$. Furthermore, you have the cellular approximation theorem which says that an arbitrary continuous map between CW complexes $X$ and $Y$ is homotopic to a (much nicer) cellular map, i.e. one which takes the $n$-skeleton of $X$ to the $n$-skeleton of $Y$. 
Because of this nice structure, if you want to prove something for all topological spaces, it's often easier to first prove it for CW-complexes and then apply the CW-approximation theorem to get it for all spaces. The CW-approximation theorem says that for any topological space $X$ there's a CW complex $Y$ and $f:Y\rightarrow X$ inducing isomorphisms on homotopy, homology, and cohomology. In particular, this expresses $X$ (up to homotopy) as a colimit of a sequence of cellular inclusions $Y_n \hookrightarrow Y$. Thus, the homotopy groups of $X$ are colimits of the homotopy groups of the $Y_n$ and $\pi_*(Y_{n+1})\rightarrow \pi_*(X)$ is an epimorphism.
I've listed several reasons above why the category $\mathcal{CW}$ of CW complexes is nice. More reasons: it contains the category of Graphs, geometric realizations of  locally finite  simplicial sets are in $\mathcal{CW}$, and you remove such monstrosities as the Long Line (which has the weak homotopy type of a point but is not contractible). If you want to find the "right" category to do homotopy theory you should first go to compactly-generated Hausdorff Spaces (CGHS). You can put the Quillen model structure on this and recover $\mathcal{CW}$ as the fibrant-cofibrant objects. For all these reasons and more, many think $Ho(\mathcal{CW})$ is the "right" place to do homotopy theory. For instance, in this category you have Brown representability, i.e. necessary and sufficient conditions for a functor $F:Ho(\mathcal{CW})^{op}\rightarrow Set$ to be representable. So this lets you understand cohomology theories by their representing objects. This representability is another nice feature that you need an honest to goodness CW complex for. See the excellent answer given by our very own OP here.
A: One should not forget that certain invariants, such as Whitehead torsion, are defined 
using a choice of CW structure. 
A: 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:


*

*CW complexes have nice point-set and homotopical properties.

*CW complexes have nice computational properties (for example a useful filtration).

*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. 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. 
