Methanol has the larger vapor pressure at room temperature because it has a lower molecular weight when compared with ethanol, which implies it has weaker intermolecular forces.
Both methanol (#"CH"_3"OH"#) and ethanol (#"CH"_3"CH"_2"OH"#) have the capacity to form hydrogen bonds because of the #"-OH"# group, so from this standpoint there is no difference between the two.
However, the difference comes when the weaker London dispersion forces come into play. Ethanol has a molecular weight of #"46 amu"#, while methanol has a molecular weight of #"32 amu"#.
This difference in molecular weight means stronger London dispersion forces in favor of the heavier molecule - ethanol - which in turn translates into a lower vapor pressure.
As a result, the molecule that exhibits weaker intermolecular forces - methanol - will have the higher vapor pressure.
Ethanol's vapor pressure at #20^@C# is #"5.95 kPa"#, while methanol's vapor pressure at the same temperature is #"12.8 kPa"#.
Check out this answer on the topic of vapor pressure and intermolecular forces: