Ethanol and dimethyl ether both have the molecular formula C2H6O. They are

Explanation

Ethanol contains a hydroxyl group while dimethyl ether contains an oxygen bridging two carbons, so the functional group itself differs and the two compounds behave completely differently, ethanol boiling at 78 degrees Celsius against minus 25 for the ether. That large difference is due to hydrogen bonding, which the ether cannot do. Aldehydes with ketones and acids with esters are the other standard pairs.