What's the difference between the center of a group and a normal subgroup?

If $x\in Z(G)$, you have that $g^{-1}xg = x$ for every $g\in G$, whereas if $H$ is normal and $x \in H$, you only have that $g^{-1}xg \in H$. This is a much weaker condition.

In other words, the center is invariant pointwise under conjugation by $G$, whereas in general normal subgroups are only invariant under conjugation as a whole subgroup.

The statements are not equivalent. What you’re missing is that $Hg=gH$ does not imply that $hg=gh$ for all $h\in H$: the set of elements $\{hg:h\in H\}$ can be equal to the set of elements $\{gh:h\in H\}$ without each of the individual products $hg$ and $gh$ being the same.

For a concrete example of this, let $G=S_3$, the symmetry group of an equilateral triangle; you can see its multiplication table here. Let $H=\{e,d,f\}$; it’s easy to check that $H$ is a subgroup of $G$. Then $aH=\{a,b,c\}=Ha$, but $ad=b\ne c=da$. You can go on to check that $xH=Hx$ for every $x\in G$, so that $H$ is normal in $G$, but none of the elements $a,b$, and $c$ commutes with $d$ or $f$.