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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.