Why is distillation not a viable way to separate ammonia from water?
Having embarrassed myself with my first answer, let me try this again. Ammonia and water hydrogen bond exceptionally well. In fact, the $\ce{O-H...N}$ value is 29 kJ/mol while the $\ce{O-H...O}$ value is 21 kJ/mol. (Ref) So, if there is any water around, ammonia would be happy to become solvated as they recondense. 

Also, ammonia is a weak base and reacts with water (which is not usually the case when you're doing a distillation). $$\ce{NH3 + H2O &lt;=&gt; NH4+ + OH-}$$ So, if there's any water around as the ammonia is distilling and they recombine, they would also react which would make it difficult to remove the last of the water. (Some reaction would occur because the lowest energy point of the equilibrium is not at reactants or products.)

So, as noted in the comments, distillations to prepare dry ammonia use a scrubber to remove the water.