Thanks for the peer assist, Youssef!
This is one of those nomenclature things. Ammonium is NH4+, and it stands alone as a cation, capable of binding to anions (as Youssef mentioned). Amino however, refers to an amino group within a compound. Amino refers to the R-NH2 and R-NH3+ group (with R being any group) that we can find in ie. amino acids.
NH3 on its own is ammonia. In organic compounds, amines refers to anything with a nitrogen, essentially (can be primary, secondary, tertiary).
Hope that helps! :)