Yes, that's right. The quote introduces a direct question with I never know:
I never know, what's the difference between a stalagmite and a stalactite?
You've turned it into an indirect question (a wh-relative clause):
I never know [ what the difference between a stalagmite and stalactite is. ]
The main difference is that the indirect question doesn't take the form of a question grammatically, so it doesn't undergo subject-auxiliary inversion, and it ends with a period rather than a question mark. The other difference is that the original sentence, containing a direct question, can contain a comma between the matrix verb and its complement; see this answer for a description of that structure.
Forget "no" and "not" for a second - they seem related, but it is a false comparison.
Instead, think about
"There is" vs. "There is not". These are related expressions but one is positive and one is negative.
Things are simple when the (reflexive) subject of the sentence is expressed positively.
For example, suppose the subject is "life".
There is (life on Mars).
There is not (life on Mars).
The complication is that sometimes the subject is expressed negatively.
Suppose the subject was "no life".
The verb-positive sentence remains the same.
There is (no life on Mars).
The verb-negative sentence has to change, though, because English (unlike Spanish) has rules against double negatives.
It is incorrect to say:
There is not (no life on Mars).
Instead, we say
There is not (any life on Mars).
Other examples of a negative subject becoming positive because the verb is negative:
There is no reason to agree with him. There is not any reason to agree with him.
There is no way out of this situation. There is not a way out of this situation.
There is no one alive who remembers him. There is not anyone alive who remembers him.
In each case, the "not" in "There is not" is part of the verb, making the verb negative, whereas the "no" is part of the subject, making the subject negative. So you can't really compare "there is no" with "there is not" - just remember that in English, you are not allowed to have both the verb and subject expressed negatively.
Best Answer
Yes, you do need the not at the end.
You've got a sort of a parallel construction going on, with some paraphrasing to make it less duplicative. You could rephrase the entire thing thus:
When you are asking about a situation with an explicit negation such as not in it, your question about it needs to match it:
In this case, your second piece of the question is a shortened form of "If [he was] not [right]" and therefore you must ask "why not?" to properly match it.