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".
possible duplicate of Difference between "not a", "no" and "not any"
– FumbleFingers Reinstate Monica – 2015-07-05T14:22:17.983@FumbleFingers, As a matter of fact I don't see any difference between my question and the one you linked but the truth is that none of the answers to that question solve my confusion whereas this answer, here, brilliantly solved my problem. So what shall I do next? – Lucian Sava – 2015-07-05T15:54:15.297
You can still upvote and accept answers if the question is a duplicate and the question will not be deleted. – Catija – 2015-07-05T16:04:26.487
Lucian, I agree absolutely agree @Kirt's answer here (which I have upvoted) addresses your question better than any on the "original" - and in fact it's probably a better "generic" answer covering both questions than any of the earlier ones (none of which I've upvoted). Catija has the right of it in terms of what you can/should do, but possibly a mod might "merge" the two questions. Or if yours does get closed, anyone (incl you) could add a link to it on the original. Or Kirt could copy and paste his answer into the original as well (I'd quite happily upvote it again! :) – FumbleFingers Reinstate Monica – 2015-07-05T16:24:32.197
The other question is being closevoted as a duplicate of this one. Shouldn't we close just one of them? – Lucky – 2015-07-05T22:45:43.840