In quantum mechanics a general state of a system is a linear superposition of basis states. Basis states are the eigenstates of the Hamiltonian. It is a complex vector space endowed with a notion of scalar product and a length element. This is known as Hilbert space also. Hilbert space is an example of a state space.

A phase space for a single particle is a six dimensional space spanned by three components of position and three components of momentum. Evolution of a dynamical system can be drawn as trajectories in phase space. For a N particle system the phase space is 6N dimensional. With time the system evolves in such a 6N dimensional phase space. There is no well defined notion of length or scalar product in the context of a phase space. So a phase space is not a vector space.

better you ask phase space vs ordinary spacetime!