Learn English – ny difference between “anyone” and “any one”

"Hence" means "from here". "Get thee hence!" is old English for "get out of here". "Henceforwards" is "from this time on".

"Thus" means "in this way". In older English, you might say "he opened the door thus", while demonstrating the action.

In logical senses, you can use them in similar contexts. "If x is true, then y is true. X is true. Hence, y is also true." "If x is true, then y is true. I have proved x is true. Thus, I have proved y to be true also."

Think of them as "based on this fact" and "in this way" and you can't go far wrong.

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).