Not much difference really. the only difference is the noun that they relate to. A bloom is usually a flower that grows on the ground, and is looked after because it is beautiful. A blossom is usually a flower on a tree, like a cherry tree. Because the flowers tend to all appear at the same time in the year, the flowers are said to blossom or bloom. If used figuratively, the verb means 'to reach a point in life where the inner beauty suddenly becomes visible, like a flower opening on it's plant'

Both can be nouns and verbs, and "blossom" is the older English word. Cognates of "bloom" literally mean "flower" in neighbouring languages. There's not really much difference.