Looking at the etymology of both adjectives sheds some light on the subtle differences in the meaning of "nötig" *vs. "notwenig":
nötig
from Old High German "nōtag", "nōtīg" = needful, having the need
notwendig
used from the 16th Century in the literal meaning "geeignet, die Not abzuwenden" = averting the need
Are "nötig" or "notwenig" interchangeable?
In most cases "notwenig" can be replaced by "nötig" in contemporary German but "nötig" can not always be replaced by "notwendig". This is mostly the case in colloquial usage, when an an idiomatic expression is involved, as has already been said in conjunction with "haben", or in a more figurative meaning.
Examples
Zwei Stunden oder, wenn nötig, auch länger
Sie braucht nötig Ruhe.
Das wäre aber nicht nötig gewesen!