I'm confused about what precisely is the difference between Grover's Algorithm and Amplitude Amplification. I've heard some sources say they are different algorithms and others say they are the same thing. As I currently understand it,
In Grover's algorithm you are searching for a state $|\omega\rangle$ (or multiple states) marked by an oracle, and you find this by applying Grover iterations that comprise of phase inversion $I-2|\omega\rangle\langle \omega|$ and mean inversion $I-2|s\rangle\langle s|$ in alternation.
In Amplitude Amplification, you have a projection operator P that projects you on to a "good" subspace, which seems to essentially boil down to marking a set of states. Then you have an operator $S_P = 1 - 2P$ that does phase inversion on those states and $S_\psi = 1 - 2|\psi\rangle\langle \psi|$ which seems to invert about your initial state $|\psi\rangle$.
So is the difference that in AA you are starting in an arbitrary state $|\psi\rangle$ and in Grover's you set $|\psi\rangle$ to be the uniform superposition state $|s\rangle$? Are there other important differences I am missing?