Difference between weak and Euclidean topology on CW complex

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Weak topology on CW complex $X$ is defined with a condition:

Subset $A \subseteq X$ is open in $X$ iff $A \cap e$ is open in $e$ for every cell $e$ in $X$.

As far as I understand, in case when CW complex is embedded in Euclidean space, the induced Euclidean topology and the weak topology do not necassary coincide on infinte CW complexes, but I can't imagine any simple example of such CW complex.

Can anyone give me an example and show why it is better to take the weak topology?

Thanks!