I'd prefer to go to the cinema. Would rather is followed by the infinitive without to. Would prefer is followed by to + infinitive or a noun. I'd rather have fruit juice.

When would you use would rather?

I would rather ('I prefer', 'I would prefer') is used as a modal auxiliary verb. It is followed by the infinitive (without 'to') when its subject is the same as the subject of the next verb. This happens when we talk about what we would prefer to do. I would rather (or I'd rather) stay with you.

What is the difference between prefer and would prefer?

Both "prefer" & "would prefer" express the preference. This site said "You can use 'prefer to (do)' or 'prefer -ing' to say what you prefer in general" & "We use 'would prefer' to say what somebody wants in a particular situation (not in general)".