I am having some trouble understanding the difference between the $p$-completion and a $p$-local space.

If $X$ a simply connected space has all higher homotopy groups finitely generated, then the $p$-local approximation $X\rightarrow L_{p}X$ coincides with the $p$-completion $X_{p}^{\wedge}$. Is this correct?

Now if $X$ is not simply connected such that the fundamental group is abelian, do we have that $ L_{p}X \simeq X_{p}^{\wedge}$?

I think it is true if $X$ is a circle, more precisely, $S^{1\wedge}_{p}\sim L_{p}S^{1}\sim K(\mathbf{Z}_{p}^{\wedge},1)$. Am I wrong ?