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Is there any difference between не надо and не нужно? If there is, what is it?
For example: A pizza delivery guy comes to my home and brings pizza. Its cost is 650 руб and I give him 700 руб. Now he wants to give me the change and I want to tell him that there is no need for that. What would you choose? (не надо / не нужно)
Нужно is slightly less casual and more bookish, so, while phrases "Ой, Вась, может не нужно тут нам голову-то морочить" is completely OK but native speaker most probably will use на надо.
On the other hand, professor most likely will not say during a lecture 'В творчестве этого поэта надо различать три этапа' but rather нужно.
There's also необходимо which is even more (sort of) high-style than нужно but understandably not used with "не" - double negation is just not a typical for Russian (so, you can use "необходимо" instead of "нужно", but not "не необходимо" instead of "не нужно").
Also, it looks like that надо is slowly but steadily overpowering "нужно" even in official usage.
So, answering specifically the last part of your question, I'd go with "Спасибо большое, сдачи не надо".
Historically, надо is similar to а past participle form (cf. English -de, Spanish -d@ and Swedish -d(e) / -t).
Нужно, in turn, represents a cluster of historically -нъ-ending words which in modern Russian are at a borderline between an adjective and a participle (cf. English driven, striken, etc).
Both forms are related to Proto-Germanic naudiz.
In my opinion, both forms are also related to German nehmen and Dutch niemen.
The word stems are preserved in modern Russian interjections:
на! = here you are! / Take it! (lit. 'you-take-it!', often with a less friendly / respective connotation) and
ну! = give it (away)! I'm waiting / I agree / come on / yep, ok (lit. 'I-take-it' or 'let me take it'), used with a somewhat more friendly emphasis.
Hence, the overtones would be like
не надо = It is not needed 'from your part / point of view / for/from you'
не нужно = it is not needed from my part / point of view / for/from me'.
But most native speakers would not care about the difference, anyway, or would not explain it.
PS The proverbial etymology of надо as a form of надобно doesn't stand any etymological scrutiny not just because words starting with на- in general were very unlucky when interpreted by Vasmer and Co, but also because such an etymology doesn't explain the sound alteration in надо and нужно.
Also, stating that -до is not a suffix but a part of stem one has to admit the same for words like чудо, стадо and говядо.
