I'll give an answer from the point of view of an algebraic topologist, somebody who cares about examples and computations. This all goes way back and has nothing to do with modern generalities. First off, calculationally, algebraic topologists can compute, like to compute, and need to compute mod p homology. It is far more accessible than p-local homology, and it is the focus of completion, not localization, at p.

An elementary concrete example of homotopy information invisible to p-localization is that $K(\mathbb Z/p^{\infty},n)$ becomes equivalent to $K(\mathbb Z_p,n+1)$ after $p$-completion. This jacks up to Quillen's equivalence between the $p$-completions of the algebraic $K$-theory space $K(\bar{\mathbb F}_q)$ and $BU$ where $p\neq q$. The homotopy groups of $K(\bar{\mathbb F}_q)$ are $\oplus_{p\neq q} \mathbb Z/p^{\infty}$ in odd degrees and $0$ in even degrees. The homotopy groups of $BU$ are $0$ in odd degrees and $\mathbb {Z}$ in even degrees. Obviously $p$-localization knows nothing whatsoever about this equivalence.

Calculationally, the classical Adams spectral sequence converges to the stable homotopy groups of the $p$-completion, not the $p$-localization, of the space (or spectrum) one starts with; that is also true unstably. The Atiyah-Segal completion theorem says that the topological $K$-theory of $BG$ is the completion of $R(G)$ at its augmentation ideal. If $G$ is a $p$-group, this is just $p$-adic completion (on the reduced level). That is a place where use of pro-groups helps calculation, but in most of the calculational applications use of pro-objects is not especially helpful.

There are tons of places where $p$-completion is essential to the proof or statement of results: the Sullivan conjecture, the Segal conjecture, the classification of $p$-compact groups, the study of atomic spaces, etc. It is $p$-completion that is relevant to all of these and many more.

A technical reason $p$-completion is so convenient is that there are no phantom maps to worry about if one restricts to nilpotent spaces of finite type. That is also true for rationalization but not for localization at $p$.

I could go on for many pages. The more algebraic topology one learns, the more frequently $p$-completion appears and the more natural it seems.

It is worth adding that we do not understand completion very well for non-nilpotent spaces, where there are several very different notions, none well understood calculationally. This deserves more study.

I'll give an answer from the point of view of an algebraic topologist, somebody who cares about examples and computations. This all goes way back and has nothing to do with modern generalities. First off, calculationally, algebraic topologists can compute, like to compute, and need to compute mod p homology. It is far more accessible than p-local homology, and it is the focus of completion, not localization, at p.

An elementary concrete example of homotopy information invisible to p-localization is that $K(\mathbb Z/p^{\infty},n)$ becomes equivalent to $K(\mathbb Z_p,n+1)$ after $p$-completion. This jacks up to Quillen's equivalence between the $p$-completions of the algebraic $K$-theory space $K(\bar{\mathbb F}_q)$ and $BU$ where $p\neq q$. The homotopy groups of $K(\bar{\mathbb F}_q)$ are $\oplus_{p\neq q} \mathbb Z/p^{\infty}$ in odd degrees and $0$ in even degrees. The homotopy groups of $BU$ are $0$ in odd degrees and $\mathbb {Z}$ in even degrees. Obviously $p$-localization knows nothing whatsoever about this equivalence.

Calculationally, the classical Adams spectral sequence converges to the stable homotopy groups of the $p$-completion, not the $p$-localization, of the space (or spectrum) one starts with; that is also true unstably. The Atiyah-Segal completion theorem says that the topological $K$-theory of $BG$ is the completion of $R(G)$ at its augmentation ideal. If $G$ is a $p$-group, this is just $p$-adic completion (on the reduced level). That is a place where use of pro-groups helps calculation, but in most of the calculational applications use of pro-objects is not especially helpful.

There are tons of places where $p$-completion is essential to the proof or statement of results: the Sullivan conjecture, the Segal conjecture, the classification of $p$-compact groups, the study of atomic spaces, etc. It is $p$-completion that is relevant to all of these and many more.

A technical reason $p$-completion is so convenient is that there are no phantom maps to worry about if one restricts to nilpotent spaces of finite type. That is also true for rationalization but not for localization at $p$.

A more sophisticated example is Mandell's algebraization of $p$-adic homotopy theory, the closest analogue we have of the classical algebraization of rational homotopy theory. (The last part of Harpaz's answer concerns this.) I could go on for many pages. The more algebraic topology one learns, the more frequently $p$-completion appears and the more natural it seems.

It is worth adding that we do not understand $p$-completion very well for non-nilpotent spaces, where there are several very different notions, none well understood calculationally. This deserves more study.