Me bateu essa dúvida após assistir um vídeo sobre o islã e fui a busca! De acordo com o Merriam-Webster:bigot
(n): a person who strongly and unfairly dislikes other people, ideas, etc. : a bigoted person; especially : a person who hates or refuses to accept the members of a particular group (such as a racial or religious group)
i.e.: He was labeled a bigot after making some offensive comments.
an incorrigible bigot who hasn't entertained a new thought in yearsfanatic
(n): marked by excessive enthusiasm and often intense uncritical devotion <they're fanatic about politics>
i.e.: Because of her fanatical views, her friends know better than to discuss religion with her
Dessa forma, me parece que bigot
encaixa mais na ideia de intolerância e rejeição às ideias de determinado(s) grupo(s), enquanto que fanatic
tem mais a ver com o entusiasmo excessivo à alguma ideologia/religião/time.