Surprisngly, these are quite different words. Both are adjectives, and they both have noun forms (fluourescence/florescence) and verb forms (fluoresce/floresce).
Fluorescent describes two kinds of things that shine or glow with their own light: (1) natural things which give off light, such as fireflies, deep sea fish, some bacteria, and some minerals; and (2) anything which seems as if it gives off light, because of its bright color:
Many creatures who live in darkness have fluorescent cells or bacteria in their bodies.
Her fluorescent yellow dress really caught my attention.
Florescent, in contrast, refers to blossoming, or flowering, like plants in the spring, or metaphorically, anything that can ‘blossom’:
Her garden became beautifully florescent in the month of May.
The florescence of science in Europe during the Englightenment was sparked by contact with the Muslim world.