Difference between quasar and Active Galactic Nuclei?

All quasars are AGN, but not all AGN are quasars.

AGN is a terminology that comes later than quasars. Quasars is the term applied at the beginning, when the first objects of this type have been discovered. They were radio-loud and point-like (the so-called quasi-stellar radio sources). This characterization still holds nowadays. Another property is that quasars are cosmological, that is they are objects in the distant universe (redshift $z > 1$).

AGN is the most general term we can use to refer to active galaxies. AGN include all of them (Seyfert galaxies, Quasars, Blazars, etc.). In the end, it seems that the difference among various types of AGN is due to a mix of angle-of-sight/epoch-of-observation.