Answer
The molecules have dramatically different boiling points because of hydrogen bonding. Explained in detail below.
Work Step by Step
Structure of diethyl ether is $CH_3-CH_2-O-CH_2-CH_3$.
Structure of butan-1-ol is $CH_3-CH_2-CH_2-CH_2-OH$.
Both these compounds have one oxygen, therefore they can both form hydrogen bonds with water. So, their solubilities are similar.
But the molecules of butan-1-ol can also hydrogen bond with each other, but the molecules of diethyl ether cannot. So, more energy is needed to vaporise butan-1-ol. So, boiling point of butan-1-ol is much higher than that of diethl ether.
