''similarites/differences between phase space and Hilbert space?''
There are no similarities except that they are both infinite-dimensional vector spaces. How to use the two is completely different. Thus there is no book that can answer that.

But if you mean the differences and similarities of classical and quantum physics, my book
Classical and Quantum Mechanics via Lie algebras
will be the right thing for you. It presents classical and quantum mechanics from a unified point of view, in which the differences (that in typical books are maximized) are minimized.

But if you want an in depth discussion of phase space and Hilbert space in QM, the right book to read is probably the book Harmonic analysis in phase space by Folland. It involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts, all at the intersection between Hilbert space techniques and phase space techniques.

This post imported from StackExchange Physics at 2014-03-22 16:59 (UCT), posted by SE-user Arnold Neumaier