Pseudo-inverses equivalence

Question

I am wondering what is the relation between Pseudo-inverses and the following:

$Ax=b$, when $A$ is singular, then $x = \left( A^T A \right)^{-1} \left(A^T b \right)$.

Then, what is the difference between $\left(A^T A\right)^{-1} A^T$ and $A^+$?