Localization and p-adic completion of Integers coincide?

No, $\mathbb{Z}_p$ is much larger than $\mathbb{Z}_{(p)}$. Indeed, $\mathbb{Z}_p$ is uncountable, since it has an element $\sum a_np^n$ for any sequence of coefficients $a_n\in\{0,1,\dots,p-1\}$. On the other hand, $\mathbb{Z}_{(p)}$ is a subring of $\mathbb{Q}$ (the rationals with denominator not divisible by $p$), so it is countable.

$\mathbb{Z}_{(p)}$ is a proper subring of $\mathbb{Z}_{p}$, and the latter one is complete discrete valuation ring, but $\mathbb{Z}_{(p)}$ is not complete (but still DVR with respect to the same valuation). However, if you take completion of $\mathbb{Z}_{(p)}$ with respect to the $p$-adic norm, you get $\mathbb{Z}_{p}$.