"No better than" vs "Not better than"
Solution 1:
Good question.
Both sentences are correct, grammatically. The former, however, is more idiomatic than the other.
As for meaning: well, they mean almost the same thing, but they sound just a little different. That is perhaps true of many (if not most) idiomatic expressions.
As for how they come into being and remain popular is simply one aspect of the evolution of language. You'd think that by subtracting just one letter--in this case, a T--wouldn't make a difference, but it does.
Perhaps expanding the context of the two sentences might prove helpful: Two people are talking about Minnie the Mooch. (A mooch is a person who takes advantage of the generosity of others but is not himself generous.) One person considers Minnie to be his friend, while the other person does not. The former thinks Minnie's habit of asking for things but not returning the favor when someone else needs something from him is simply part of who Minnie is. He accepts him as he is.
The latter person, on the other hand, thinks Minnie behaves like a low-life beggar. He might then say,
"Minnie is no better than a beggar."
This comment--again, an idiomatic expression--reveals the contempt this person has for Minnie much better than
"Minnie is not better than a beggar,"
which is not idiomatic and would not have the same impact, force, or vehemence as the idiomatic expression.
As for why this is: well, that again is how language evolves.
In conclusion, let's take a sentence from my answer, above, and change a few words in it. The two sentences say almost the same thing, but not quite, because the first sentence contains an idiom while the second sentence does not.
Perhaps expanding the context of the two sentences might prove helpful.
Perhaps expanding the context of the two sentences will be helpful.
Can you tell which sentence contains the idiom? If you guess the first sentence, you are correct. "Might prove helpful" is an idiomatic way of saying "will be helpful."