There is no verb "yukimasu" that means to snow. Only "yuki ga furimasu" means "it snows / it will snow".
Someone correct me if im wrong but i think ゆきます is very vague because there is no topic marker. The translation would be like "snow, it (something else?) exists" the particle が is used to indicate we are definitely talking about snow as the main topic and ふります means "it falls" so ゆきがふります more literally means "snow falls" or translating to english "it snows" because english likes to verbify nouns.
Someone please correct me if I am wrong, but I believe the main difference is that ゆき means snow when used as a noun. As in ゆきがあります - "There is snow" or ゆきです - "It is snow" so you would need to include a verb like ふる in order to say "snow is falling." I am not sure what the verb is for "to snow" ゆきます, I am less familiar with unfortunately, but I find it in the form of 征きます、meaning "to conquer."