It seems the definition of the center of a group and a normal subgroup are the same so I'm wondering what the difference is between the two?
A group $H$ is normal in $G$ iff $Hg=gH$ for all $g \in G$.
The center of a group $Z(G) = \{z| \in G$ and for all $g \in G, gz=zg\}$
Those statements seem equivalent to me.