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Consider these two examples:
神｛かみ｝を信｛しん｝じるの？
and
神を信じますか？
To me, both mean 'Do you believe in God/gods?', with the first example possibly slightly more casual than the second. Other than that, are there any other nuances? I can't really give a context, but in general they seem to be used slightly differently and I can't seem to pinpoint the exact differences.
If I had to make タメ語-丁寧語 correspondences, I would say that roughly
神を信じる？ corresponds to 神を信じますか？
and
神を信じるの？ corresponds to 神を信じるんですか？
The versions without の are plain yes/no questions, something you ask when there is nothing in the discourse/environment hinting either way.
The versions without の are used when something in the discourse/environment has provided a hint to the speaker that the addressee believes in God (possibly contrary to previous expectation).
TBH, I am a little perplexed by what appears to be the sudden popularity of the question-ending 「の」 among Japanese-learners in the last several years. I am perplexed because it is used incorrectly or in an unnatural way at least 60-70% of the time by them.
While one could claim that 「神を信じるの？」 and 「神を信じますか？」 have the same surface meaning, the intended nuances would be quite different if said by native speakers. Those two sentences would not be interchangeable in real life by our standards.
「神を信じますか？」 is plain and simple. It means "Do you believe in God?" with no nuance intended. The speaker does not expect a particular answer.
「神を信じるの？」 is not plain in that the speaker already sort of expects a nagative answer. Someone who strongly believes in God himself would NOT say this.
OP is correct regarding the informality of the 「の」.
