Why does NH3 have a lower boiling point than H2O?

As you know, the normal boiling points of water, #100# #""^@C#, and ammonia, #-33.3# #""^@C#..are EXCESSIVELY HIGH....especially given these are molecules of paltry molecular mass, the which gives rise to negligible dispersion force. So why so? Clearly, there is an intermolecular force operating between the water and ammonia molecules, the which you have already identified.

Hydrogen-bonding occurs when hydrogen is bound to a STRONGLY electronegative element, i.e. #"nitrogen, or oxygen,"# #"or fluorine"#...and in fact we could recognize that the boiling point of #HF#, #19.5# #""^@C# is ALSO rather high...

Now for water, the hydrogen bonding network is particularly extensive...#H-stackrel(-delta)O-stackrel(+delta)H#, and extends INTERMOLECULARLY...

Whereas for ammonia....

...given that the nitrogen ATOM has ONLY the one lone pair..the opportunity for intermolecular interaction via hydrogen-bonding is diminished with respect to water. This is reinforced by the polarity of an #O-H# bond versus an #N-H# bond...i.e. nitrogen IS LESS electronegative than oxygen, and polarity is not pronounced in the ammonia molecule as compared to the water molecule...

And the end result? Well, we try to account for the experimental results....but certainly the normal boiling point is an excellent parameter with which to assess the intermolecular force.