In the sentence:
We all wish the snow would stay forever.
Why can't we say
We all wish the snow stayed forever.
What are differences between these sentences?
According to Swan's Practical English Usage, Unit 630.4, we can use past tenses in a that-clause after wish:
I wish tomorrow was Sunday.
Note that here the past tense is used to refer to the future. If we want a sentence of this structure to refer to the past, we use past perfect tenses, according to Swan:
I wish you hadn't said that.
In Quirk's Comprehensive Grammar, Unit 14.23 says that "in subordinate clauses following I wish (that), the hypothetical past may be replaced by hypothetical would or by a to-infinitive":
I wish she visited me more often.
I wish she would visit me more often.
I wish her to visit me more often.
This means we can also say:
We all wish the snow to stay forever.
(Native speakers might find this sentence unnatural. It looks unnatural to me.)
I went on reading Swan further and found one interesting distinction (Unit 630.5): Wish ... wouldn't sentences can be used as orders or requests:
I wish you wouldn't drink so much. (meaning, "Please don't drink so much".)
I wish you didn't drink so much. (here, the meaning is closer to "I'm sorry that you drink so much".)
So, in such negative sentences the use of the simple past gives a different shade of meaning compared with the use of the infinitive with the modal would.