A geek likes something. A nerd likes something too much. So Sheldon is a nerd, because he enjoys science at the expense of other things such as relationships.

They're basically the same thing now; people use them interchangeably.

What I noticed is that when people say "nerd" they refer to intelligence and when they say "geek" they refer to appearance.

This is how it is in London, I'm not sure how it works anywhere else.

Nerd to be a full-fledged genius is incorrect. Due to the often technical nature of their speciality nerds are more likely to be of relatively high intelligence while Geeks are a more diverse group in terms of intellectual ability. Source: https://www.wikihow.com/Tell-the-Difference-Between-Nerds-and-Geeks

I would say he is both. There is overlap in the meanings.

Opinions vary: mine is exactly the reverse of Philip's. There is a good article here https://www.dictionary.com/e/dork-dweeb-nerd-geek-oh/ which looks at the overlapping terms geek, nerd, dweeb and dork in American English. They have precise origins in high school culture and its presentation in film and TV, giving a false sense of fixity to their use, which has spread far beyond their original context. Most people are unfamiliar with American youth tribes and their names, and live in a different world from them. Dictionaries such as the OED reflect this: the definitions are so broad as to overlap completely. Part of the problem is that these terms are value-laden, and carry different emotional baggage for different people. For example, some non-geeky, non-nerdy people use these terms as insults, others as oddly admiring terms for internet billionaires etc. Some people with geeky/nerdy traits proudly identify as geeks or nerds, others are deeply offended to be branded in this way. Different people use these terms for different purposes, and they stretch the meaning to fit. This gives you the freedom to do the same or, if you are a word nerd (or geek) like me, simply avoid using them.