A comparison between the structures "would rather" and "would prefer"

I know that "i'd rather", like "I'd prefer" is used for specific situation and not general ones. On the other hand I know that the second sentence below doesn't work here. But the question ins that why the second sentence doesn't work here?

1 - I’d prefer coffee to tea tonight.

2 - I’d rather coffee to tea tonight.

My next question is that if "I'd rather" is in common use in AE.

Answer

prefer indicates a choice of things (nouns), while rather indicates a choice of actions (verbs).
Prefer is a tr. verb and rather is an adverb.

So using your examples,

1 - I’d prefer coffee to tea tonight.

this is OK because it is used as a tr. verb.

2 - I'd rather have coffee to tea tonight.

Since rather is an adverb, a verb (both to be modified by the adverb, and just to have one in the sentence), is needed. In this case, have.