When I put both of them in my dictionary I got the same definition, so it is clear that they are the same.
I thought "nonetheless" is more formal than "nevertheless"
Nonetheless vs. Nevertheless
The terms ‘Nonetheless’ and ‘Nevertheless’ implicate the same meaning. However, the usage of both words is drastically different owing to the sentiments of the people. According to the general consensus, people like using the word ‘nevertheless’ instead of ‘nonetheless’ simply because one sounds better than the other.
]Although COCA confirms that 'nevertheless' (with 13,981 citations) is more commonly used than 'nonetheless' (9,795), there is no evidence to show that this is because 'one sounds better than the other'.