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I am the senior New Testament editor. I like the NASB and used it for many, many years. But its fundamental flaw is that it is too literal. Often the translators would pick the first entry out of BAGD and use that for the word all the way through. Take, for example, polis. NASB translates it "city" all 162 times. Yet polis is either a city or a town. Since Koine Greek had no good work for town, polis covered a multitude of sins. The NET translators considered size, relation to government, whether it was a free or bound city, etc. Several factors allowed us to come to more nuanced decisions, rather than thinking that one size fits all.
A second area where they are different is this: In the synoptic problem, NET rigorously compared the parallels. We color-coded the Greek synopsis, showing where verbatim parallels were. We felt that unless the context was completely different, we really did not have a right to translate the two passages differently.This process took hundreds of hours but has produced what I believe is the only Bible to take the synoptic parallels seriously. For example, was it a young girl, a servant girl, or a slave girl who taddled on Peter at his betrayals? The Greek is the same each time, but the translation is different. There is no reason to be cute in a translation if it's misleading. But I think the real reason was simply that the NASB editors didn't consult the parallals rigorously.
A third area is in the decidedly evangelical bent. This is a plague of many evangelical translations. Some phrases are rendered to fit with evangelical theology even though the Greek or Hebrew does not really yield in that direction. The ESV does this too. I wrote a couple of critiques about other translations on this score, posted at bible.org.
There are also archaisms that most scholars have abandoned long ago. Such as 'only begotten' in John 1.18; 3.16.
Finally, the translation philosophy of the NET was faithfulness to the original meaning, while NASB's philosophy was more faithfulness to the original forms. That can end up producing a misleading translation. The NET is able to both because we have more extensive footnotes than any other Bible in history. We usually give the more idiomatic translation in the text, the more literal in the notes. And the translation is not wooden.
Those are a few of the differences. I have been a consultant for at least five translations. Far and away, the NET is the best in my view.