5 Fixed my broken links. I'm not sure how they got broken...something to do with copy and paste not playing nicely with parentheses

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.

4 added 18 characters in body

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.

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 who 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."
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 because 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 $Y$. So
Because of this nice structure, if you want to prove a theorem something for all topological spaces, it's often easier to first prove it for CW-complexes and then apply the CW-approximation theorem to say 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. This 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 often enough 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 homotopy type of a point but is not contractible). If you want to get find the theorem "right" category to hold for 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 spaces once these reasons and more, many think $Ho(\mathcal{CW})$ is the "right" place to do homotopy theory. For instance, in this category you know it 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 complexescomplex for. See the excellent answer given by our very own OP here.