Generally, famous refers to recognition, and popular refers to reputation (of a person) or frequency (many people use a thing or visit a place).
Famous often, but not necessarily, has positive connotations.
Knowledge of the famous person, thing or event is widespread.
Popular usually has positive connotations, or when referring to a place, it means it is visited by many people.
The person, place or thing is largely well regarded among the people who know of it, but doesn't necessarily mean a lot of people know of it.
Perhaps it's easier understood with an example.
A famous restaurant: Is well known.
The name of the restaurant is recognized, perhaps because a well-known chef founded it, or it advertises a lot, or a well-known event occurred on the premises.
A popular restaurant:
It's usually busy.
The majority of people enjoy eating there.
Or maybe they don't, but they go there because the food is cheap, or it's the "place to be.
"
Outside of the people that eat there, it may or may not be well known.
A restaurant can be both famous and popular (everyone knows about the restaurant, and many of them go there), or popular but not famous (only the people that frequent the restaurant know of it), or famous but not popular (making it infamous ).
english.stackexchange.com/users/1871/ghoppe
Famous means well known.
Popular means well-liked.
Something or someone that is well known, but not popular could be infamous, which isn't a lack of being known, but being known for a bad reputation.
english.stackexchange.com/users/3287/Spare Oom
Famous people are likely to be popular but it isn't necessary, e.g. George Bush.
Popularity does not necessarily imply fame either.
e.g. "You are quite popular in your class."
just means liked by many. '
Famous' cannot be used interchangably here.
english.stackexchange.com/users/669/ottodidakt
A famous person is someone a lot of people know about.
A popular person is someone who is liked by people who know that person.
english.stackexchange.com/users/3279/Larry Coleman
Famous means known by many people because of their achievements or works, while popular means liked by many people.
Suppose in a school there is a very beautiful girl who is always turning heads, and there is a very smart girl who is always topping the exams.
The former is likely to be popular while the latter is likely to be famous .
english.stackexchange.com/users/None/
If a resturant is found to be contaminated by E Coli or have particularly bad food, it would become very famous, but not very popular.
english.stackexchange.com/users/4116/Daniel
Famous = "well-known".
Popular = "well-liked [main factor] and well-known".
An example: Rebecca Black is famous, but not popular.
english.stackexchange.com/users/36776/Sophia goh
Is there any difference between  famous  and  popular ?
Is there any difference between "famous" and "popular"?
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<word-choice><differences><adjectives>
english.stackexchange.com/questions/8022
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In general, the word "famous" refers to recognition whilst "popular" refers to reputation. Being "popular" usually has positive connotations whereas "famous" does not necessarily have to be positive. Famous is well-known. Popular is well-liked.
Generally,  famous  refers to recognition or to someone that a lot of people know about, whereas  popular  refers to reputation or to someone who is liked by those who know him/her. In short, famous means well known, and popular means well-liked. "Famous" generally refers to someone or something that is well-known, whereas "popular" generally refers to someone or something that is well-liked.
Outside of the people that eat there, it may or may not be well known.
Famous refers to being well-known due to achievements, or works. Popular refers to being well-liked Famous means well-known  Popular means well-liked 
"Famous" generally refers to someone or something that is well-known, whereas "popular" generally refers to someone or something that is well-liked.