Ammonia will diffuse faster if both gases are at equal temperatures.
The short explanation is that at equal temperatures both gases have equal kinetic energies. For the kinetic energy of the gases to be the same, the lighter (lower mass) gas particles must be moving more quickly.
The molar mass of ammonia is: 17.04
The molar mass of HCl is: 36.46
Faster moving gas particles will diffuse (spread out) more quickly.
This video has a more detailed discussion of this topic and Graham's law.
Hope this helps!