Generally I would expect something that is an "allergen reducer" to be able to remove very small particles from the air that typically cause problems for allergy sufferers.

An "air purifier" could be something as simple as a fan and a foam filter that removes some dust from the air and perhaps little else.

Best to check out industry standard terms such as if it is HEPA rated which is a standard. There are also measurements such as the micron size that the filter can remove from the air too that you can use as a benchmark.

Try consumer reports magazine too or other impartial research sources to compare how these devices perform.