In brief, ammonites are animal fossils, ammolites are gemstones.

Ammonites are fossilized shells of ancient cephalopods (nowadays nautilus as closest living relatives), all over our planet earth. 

For ammolites, they are also fossilized ammonite shells, with aragonite and other mineral impurities during fossilization. This transform the shell into iridescent, opal-like gemstone with awesome colors. Gem-quality ammolite can be found in southern Alberta, Canada. 

In 1981, World Jewellery Confederation (CIBJO) recognized ammolite as an organic gemstone. With iridescent colors on thin layer of fossilized ammonite shells.