Both "no one" and "nobody" are largely interchangeable to mean no person or not a single person.
No one could crack the case of the missing jewels. vs Nobody could crack the case of the missing jewels.
Both "No one" and "Nobody" function as pronouns, referring to that fact that everyone involved was not able to solve the case, which also means that no person could solve it.
"No one" is more commonly used in formal writing.
There is, however, a difference if you were to use "nobody" as a noun.
He is just a nobody (here, nobody is a noun referring to someone insignificant, and the article (i.e. a) qualifies the noun (i.e. nobody).
He is just a no one (incorrect as we do not use an article (i.e. a) to qualify a pronoun (i.e. no one)).
I hope this helps.