Visualising regular CW complex

By and large, lack of regularity is for convienience. The "standard" CW-decomposition of a 3-dimensional lens space $L_{p,q}$ has one 0-cell, one 1-cell, one 2-cell and one 3-cell. But it's impossible to make such a simple CW-decomposition into a regular one, since $H_1 L_{p,q} \simeq \mathbb Z_p$. A regular CW-decomposition with one cell in every dimension has $H_1$ free abelian.

Of course, the lens space has a regular CW-decomposition, but it's more work and more fuss to find it. This is much like how every manifold has a triangulation but you maybe don't want to work with a triangulation. The cellular boundary "degree term" is simpler, but there's far more cells, so the benefit of having a simple degree term is killed by having a complicated chain complex.

Presumably there are spaces that have non-regular CW-decompositions and lack regular CW-decompositions. But this is very much a fussy point-set topological curiosity -- the real reason one cares about regular vs. non-regular is the one given above. I think an example of a space where there is a CW-decomposition but no regular decomposition would be the interval $[0,1]$ attach a 2-cell, where the attaching map $f : S^1 \to [0,1]$ is given by:

write $z \in S^1$ as $z=e^{i\theta}$ with $\theta \in [0,2\pi]$.

then $f(z) = (\theta/2\pi) |\sin((2\pi)^2/\theta)|$

A little argument tells you if there was a regular CW-structure then there would have to be infinitely-many cells. But then you can argue this space does not have the weak topology of such a complex. Anyhow, something like that should work.