gyeoureneun nuni manhi wayo.

Adding -는 to 겨울에 (making 겨울에는) turns the time expression into a topic with a hint of contrast or generalization.

• 겨울에 simply means “in winter” (just a time marker).
• 겨울에는 means “as for winter” or “when it comes to winter,” implying you’re talking about winters in general or comparing winter with other seasons.

The particle -이 marks 눈 as the grammatical subject of the sentence.

• 눈 is what’s “doing” the action (in Korean we say precipitation “comes”).
• So 눈이 와요 literally “snow comes,” and -이 signals that 눈 is the subject.

Both are correct, but:

• 눈이 와요 is the everyday, neutral way to say “it’s snowing.” In Korean the conceptualization is that precipitation “comes.”
• 눈이 내려요 (from 내리다) also means “snow falls” and is more literary or descriptive. In casual speech, 눈이 와요 is far more common.

많이 is an adverb modifying the verb 와요, so it means “come a lot,” i.e. “a lot of snow comes.”

• 눈이 많아요 would mean “there is a lot of snow” (existence), which is OK but shifts focus from the action of falling to the state of having snow accumulated.

와요 is the present tense polite informal form (반말 → 존댓말 crossover). It’s appropriate among strangers or in everyday polite conversation.

• For more formal contexts (e.g., announcements), you’d use 와요’s stiff counterpart 옵니다.
• For casual speech with close friends or younger people, you might drop 존댓말 and say 와 (informal).

Korean is relatively flexible, but time-topic phrases like 겨울에는 usually come first for clarity. You could say:

• 눈이 겨울에 많이 와요.
  This is grammatically fine but slightly less natural, since you’d lose the topical/contrast nuance that -에는 gives.