My question is in the title. I do not really understand why water is not a superfluid. Maybe I make a mistake but the fact that water is not suprfluid comes from the fact that the elementary excitations have a parabolic dispersion curve but for me the question remain. An equivalent way to ask it is: why superfluid helium is described by Gross-Pitaevsky equation and it is not the case for water?
You refer to the Landau criterion for superfluidity (there is a separate question whether this is really the best way to think about superfluids, and whether the Landau criterion is necessary and/or sufficient). In a superfluid the low energy excitations are phonons, the dispersion relation is linear $E_p\sim c p$, and the critical velocity is non-zero. In water the degrees of freedom are water molecules, the dispersion relation is quadratic, $E_p\sim p^2/(2m)$, and the critical velocity is zero.
The Gross-Pitaevskii equation applies (approximately) to Helium, because in the superfluid phase there is a single particle state which is macroscopically occupied. The GP equation describes the time evolution of the corresponding wave function. In water there are no macroscopically occupied states. You can try to solve the full many-body Schroedinger equation, but at least approximately this problem reduces to cassical kinetic theory.
I think the best criterion for superfluidity is irrotational flow: The non-classical moment of inertia, quantization of circulation, and persistent flow in a ring. Again, these don't appear in water because there is no spontaneous symmetry breaking, and no macroscopically occupied state.
Because water is liquid at much too high a temperature. Helium is only superfluid near absolute zero. To have a superfluid, you need the quantum wavelength of the atoms given the environmental decoherence to be longer than the separation between the atoms, so they can coherently come together.