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A uniform pro-$p$ group is an inverse limit of finite $p$-groups, a Lie group, and (the restriction to $\mathbb{Z}_p$ of) a finite-dimensional Lie algebra all at the same time. I think you'd be hard-pressed to see all these connections just by looking at a system of finite groups.

There are a lot of results in profinite group theory that say "$G$ has a subgroup of finite index such that..." without necessarily giving any way to bound the index. You can't see these properties by looking at individual finite images.