- Why don't you join a monastery?
- Why do you not join a monastery?
What's the difference between the two sentences and when do I use each of them?
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To my mind, the first implies that the monastery-joiner (let's call them) has not considered joining a monastery before, and the question is a suggestion to that effect. This means that it is a rhetorical question; the monastery-joiner is not expected to answer, only to consider the possibility of joining.
The second is a more direct question. It seems to imply that the monastery-joiner has already considered joining, but has not. The question asked directly for the reason of this.
Gramatically, they have very similar meanings, and indeed the first question could be used as a direct question as well. The contraction of 'don't' works better with the rhetorical meaning in a way that 'do you not' fails to do.
As for usage... it depends on context, but the above is a good guide.