The difference is indeed hydrogen bonding. The melting and boiling points of substances are due to the intermolecular attractions between the molecules.H2O and H2S are both bent molecules and are therefore polar. And while H2S molecules are attracted by dipole-dipole interactions, water molecules are attracted by hydrogen bonds which in this case are much stronger.
Hydrogen bonding only occurs when hydrogen is covalently bonded to a small, highly electronegative element. These are N, O and F. The N, O, or F distort the bond to hydrogen sufficiently so that H can form a weakly covalent bond with another N, O or F atom in an adjacent molecule. This is the hydrogen bond, the attraction between molecules.
Since the attraction between molecules is stronger in water than in H2S, water will have higher melting and boiling points.