I've been seeing nonetheless used next to but. For example:
"... smaller but nonetheless important research programs." (From Merriam Webster).
There are possible risks, but nonetheless, we feel it’s a sound investment. (from Cambridge Dictionary)
My understanding is that "but" and "nonetheless" are synonymous and mean the same thing, although I'm not quite sure they mean the same thing in the above sentences.
Does the following example work?
She is sad but nonetheless beautiful
How do I analyse the above sentences, please? Do "but" and "nonetheless" have different meanings in these sentences? I would also appreciate if someone told me how I would punctuate sentences having "but" and "nonetheless" used like that.