What is it that is "symmetry-adapted" about symmetry-adapted perturbation theory?
The "symmetry" in symmetry-adapted perturbation theory refers to the anti-symmetry of the wave function with respect to electron exchange:

$$\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\mathbf{r}_1) $$

The most usual way to enforce antisymmetry in electronic structure theory is to enforce orthogonality between spin-orbitals.  So SAPT is intermolecular perturbation theory that includes terms that depend on the overlap between orbitals in different molecules.

Intermolecular perturbation theory gives rise to concepts like eletrostatic interactions, polarization, and dispersion.  With SAPT you then get additional concepts like exchange repulsion (aka steric or Pauli repulsion).