Intermolecular interactions between molecules of water are stronger than those between molecules of methane.
Molecules of water form hydrogen bondings and experience relatively strong intermolecular interactions. Water molecules contain [tex]\text{O}-\text{H}[/tex] bonds with hydrogen atoms bonded directly to oxygen atoms. Oxygen atoms are highly electronegative and in a hydrogen-oxygen bond the oxygen atom would take strong control over the electron pair it shares with the hydrogen atom. The hydrogen atom would be thus demonstrate a strong partial-positive charge and is capable of strongly attracting lone pairs of electrons on oxygen atoms of neighboring water molecules.
Methane molecules are nonpolar and experience intermolecular forces weaker than those found in water. Atoms of carbon are less electronegative than those of oxygen. Carbon-hydrogen bonds in molecules of methane are thus weakly-polarized and there's little incentive for hydrogen atoms to actively interact neighboring molecules. The four hydrogen-carbon bonds in a methane molecule are equally-weak in polarity and are arranged symmetrically in a tetrahedral geometry. Molecules of methane are therefore nonpolar. The only intermolecular interaction found in methane is induced dipole, when random turbulences shift electrons to one side of methane molecules, such that they start to attract each other momentarily as the dipole propagate across the molecules. The interaction produced is relatively weak as methane contain a small number of electrons.
Boiling a substance require overcoming intermolecular interactions and taking its particles apart. Raising the temperature of a sample of fixed mass increase the average energy of particles it comprises. Breaking hydrogen bonds between molecules of water takes more energy than overcoming the transient induced dipole in an equal amount of methane. Therefore it takes more energy to boil the same amount of water than equal amount of methane.
