Well explained Clive and Peter.
So going back to the original question and assuming a mono camera, the narrower filters will allow for capturing fainter details.
There are no miracles though, and even 3nm filters will perform better from a dark site and without the moon, but it is possible to capture reasonable data with 3nm filters with the moon up and from a light polluted area. Data captured through 6nm filters will be more strongly affected by the moon and by light pollution, and then 12 nm even more so.
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