Normalizer vs Centralizer In Group Theory, what is the difference between a normalizer and a centralizer of a set S? I'm having a bit of difficulty understanding it...
Thanks in advance!

There is a difference between fixing $S$ elementwise and preserving $S$ while possibly permuting its elements — you may have come across this distinction before, in linear algebra. To be precise, the centralizer of $S$ in an ambient group $G$ is
$$
Z_S = \{x \in G : \text{for each } y \in S,\, xyx^{-1} = y\}
$$
and the normalizer is
$$
N_S = \{x \in G : xSx^{-1} = S\},
$$
where $xSx^{-1} = \{xyx^{-1} : y \in S\}$. Certainly $Z_S \subset N_S$, but this containment of subgroups may be strict. For a simple example, take $G = S_3$ and $S = A_3 = \{e, [123], [132]\}$. Here $Z_S = S$, but $N_S = G$.
