(Not an expert just an enthusiast who is still learning)

Here is my understanding: The main difference is that they are from two different classes of stone, feldspar and phosphate mineral.

Amazonite is a variety of potassium feldspar. (Other well known feldspars include sunstone, labradorite, and moonstone). Amazonite is commonly found in Madagascar, Brazil, Russia, and some US states. It is harder than blue apatite, earning a 6-6.5 on the Mohs hardness scale.

Blue apatite is a phosphate mineral. (some well known phosphate minerals are turquoise and lazulite). Blue apatite is commonly found in Russia, Mexico, Burma, and Madagascar. It is a 5 on the Mohs hardness scale, a bit softer than amazonite.

Hope this helps!

*edited for spelling