I agree with Brian's comments on identical semantics and commonality of usage.
To reiterate, at least in US vernacular, there's no difference between:
a) no more and not more
b) no less and not less
As Brandin stated in his comment, both "I have no more than 1000" and "I have not more 1000" mean "I have 1000 or less." Both "I have no less than 1000" and "I have not less than 1000" mean "I have 1000 or more."
Also, Brian is correct that using "no more (less)" is much more common than using "not more (less)". I would caution you against using "not" instead of "no" with less/more. Even though it might be technically correct, I wouldn't consider it idiomatic. To me, it would sound really strange if someone used "not more (less)" instead of "no more (less)" in this case.
As for positivity and negativity, I agree with Brandin that there's an ambiguity here and that it seems that the positivity/negativity is more mathematical than it is related to connotation.
Furthermore, I disagree with your book's translation for statement number 2. "I have no more than 1000," does not mean "I have only 1000." "I have only 1000" means "I have exactly 1000." As stated previously, "I have no more than 1000," means "I have 1000 or less."
Regarding the negative connotation of only in this case, I think there's some ambiguity due to both a lack of context and a tendency for English speakers to use "only" as a modifier incorrectly. Only isn't exclusively negative, and whether you can say a sentence with "only" has a negative connotation depends more on the placement of the word only, the topic, and who's saying the statement with "only" in it.
Consider the following statements:
1) "I have 1000 dollars"
2) "I only have 1000 dollars"
3) "I have only 1000 dollars"
4) "I have 1000 dollars only"
Statement 1 is neutral and it expresses no limitations. The person has $1000 or more. A millionaire has $1000. They also have at least $1,000,000.
Statement 2 expresses a limitation and is negative. Since "only" is before the verb "have," the limitation is related to how much the person has. The person has exactly 1,000 and no more. The "only have" implies that there is alternative kind of action or state. In this case, it's having more. However, the implied negativity is related to the topic (money), not the structure of the sentence.
To elaborate, only + verb, expresses that there is an alternative action state. The negativity associated with each statement varies depending on the topic. It also depends on who you imagine saying it.
2a) "He only came to the party to eat cake."
2b) "She only went to the store."
2c) "I only eat cake."
2d) "She only raises cows."
2e) "They only travel by private jet."
Regarding 2a, imagine throwing a party to find out that someone came just to eat your food. In that scenario, you probably wouldn't feel positively if you found out someone only came to your party to eat cake. The alternative state/action here is coming to the party for a different (hopefully better) reason, or not going to the party at all. I'm sure there's an example that could make this statement positive, but I can't think of a good one. **
Regarding 2b, this could be a defense or a negative remark. If your sister is implicated in a murder trial, you might be inclined to say, "She only went to the store!" which is a positive alternative to, "She went to the store and the committed an act of murder." If you paid or expected someone to help you with a few errands, you might be disappointed and exclaim, "He only went to the store!"
I'm not going to belabor this point anymore.
So, regarding statements 3 (the one from your book) and 4, since the modifier "only" isn't immediately before the verb "have", we know for certain that there's a limitation; the person has exactly 1,000 dollars. We do not know, however, whether the limitation extends to the verb "has" because no alternative action is explicitly implied by the syntax (the "only" doesn't appear immediately before the verb). Therefore, you would need more context beyond just the topic of the sentence (money) and who is saying it to make a good guess about full meaning of statements 3 & 4; They could mean only "have exactly", which is neutral, or "only have," which has a negative when talking about money.
So, in conclusion, had your book's translation been "I only have 1000 yen," then you could be confident in saying that's negative, but it still would not be a correct translation of "I have no more than 1000 yen." The positivity/negativity of the translation must have been misinterpreted to mean connotation, but the positivity/negativity only relates to relative amounts (like greater than/less than).
**I don't want to confuse anyone with a misplaced joke, but I wanted to write "It's my party, I'll example if I want too."