- Would you stay here or go home?
- Would you rather stay here or go home?
Do those sentence make a difference?
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These sentences are indeed different. The first sentence asks about an action the second person would take under unspecified conditions. It deals with a hypothetical situation. To resolve ambiguity/put it in context, I would expect it to be preceded by another phrase, probably starting with "if". For example, say we are planning an outing to the zoo for some future date. Taking the weather forecast into account, I would say:
If it rains that day, would you stay home or still go to the zoo?
The second sentence, definitely the more natural and idiomatic of the two, asks specifically about the second person's preference. This is the function of the word "rather", to evoke specifically wanting or desire. To put the second sentence in context, pretend we are already at the zoo and it starts raining, I would say:
Oh, it's raining. Would you rather stay here or go home?