In the original quote (here with my emphasis):
In psychology, jamais vu (/ˈʒɑːmeɪ ˈvuː/; from French, meaning "never
seen") is the phenomenon of experiencing a situation that one
recognizes in some fashion, but that nonetheless seems very
unfamiliar.
nonetheless explicitly sets two roughly opposite ideas against one another:
one recognizes in some fashion ... seems very unfamiliar
If you leave out nonetheless, you are lessening the speaker's explicit acknowledgement of their oppositeness.
Consider:
1. I have drunk a glass of water. I feel thirsty.
2. I have drunk a glass of water but I feel thirsty.
3. I have drunk a glass of water. Nonetheless I feel thirsty.
4. I have drunk a glass of water but nonetheless I feel thirsty.
In #1, there is no explicit connection of the two stated facts.
In #2, but explicitly connects the two stated facts.
In #3, nonetheless explicitly connects the two stated facts.
In #4, but explicitly connects the two stated facts and nonetheless reinforces the explicit connection.
nonetheless can be moved to the end of the sentence, punctuating the idea:
I have drunk a glass of water but I feel thirsty nonetheless.
Nonetheless means "not any less (here, thirsty) [for having done so]"
We can also use the word "still", which means "continue to".
I have drunk a glass of water but I still feel thirsty.
I have drunk a glass of water but I continue to feel thirsty.