I thought I'd join in and give a few thoughts on this topic, spurred on by Patrick's stream of the same name.
In contrast to Patrick, I'd never heard of John Piper before I heard about the controversy between him and Tom Wright.
I've not read John Piper's book against Wright but the impression I got right from the start of Wright's book-in-reply is that he was somewhat frustrated at being misrepresented.
That Piper (and others) directed their criticism specifically at Wright is presumably because his books relating to the New Perspective have achieved popularity outside academia. I don't know how much of Piper's criticism applies to the New Perspective in general versus Wright's specific views (which aren't necessarily the same as, say, Sanders or Dunn).
The New Perspective came out of a shift in understanding Paul's writings in the context of Second Temple Judaism.
In particular, some scholars felt that many of the Lutheran and Reformed views interpret Paul too much in the context of 16th-century Roman Catholicism rather than 1st-century Judaism. In other words, they project on to Paul's opponents too much of their own opponents.
So the New Perspective is firstly an attempt to better understand what 1st-century Jews thought and hence what Paul's issue with them might be.
The idea is that if we better understand Second Temple Judaism, we'll better understand what Paul was arguing against and if we better understand what Paul was arguing against, we better understand Paul.
It would seem to me that this is worth pursuing, even if there may be some disagreement about what we find along the way.
So it is odd to me when people suggest it's not even a path worth pursuing (or is dangerous to pursue).
One interesting exercise is to read Galatians and ask yourself what Paul is referring to by "works of the Law".
And in Galatians 5–6 in particular, what does Paul tell the Galatians to do and what does he tell them not to do?
Another interesting exercise is consider what Paul says in Galatians 3 about Abraham and his readers' relationship to him and why that relationship exists.
It is a challenge when a word takes on a different meaning in our larger discourse than what it does in the biblical text.
For example, the word "church" nowadays has a lot of meaning not covered by the word ekklēsia. It is dangerous therefore to read all senses of "church" into the word when we see it in the text.
Similarly with "justification".
Alister McGrath, in his wonderful book Iustitia Dei: A History of the Christian Doctrine of Justification, says the following (also quoted by Wright):
The doctrine of justification has come to develop a meaning quite independent of its biblical origins, and concerns the means by which man's relationship to God is established. The church has chosen to subsume its discussion of the reconciliation of man to God under the aegis of justification, thereby giving the concept an emphasis quite absent in the New Testament. The "doctrine of justification" has come to bear a meaning within dogmatic theology which is quite independent of its Pauline origins.
There is nothing wrong with using the word "justification" in this sense but we must be careful reading this sense into Paul's use of it (or into our translation of dikaiōsis and cognates).
It's also a challenge that words like just/justify/justification are cognate with right/righteous/righteousness in Greek but not in English.
But perhaps the key ambiguity in the New Perspective debate is that pistis can be translated as "faith" or "faithfulness" and in particular pistis Christou can be translated , "faith in Christ" vs "faithfulness of Christ".
Compare, for example, Romans 3.22 in ESV:
the righteousness of God through faith in Jesus Christ for all who believe.
vs NET:
the righteousness of God through the faithfulness of Jesus Christ for all who believe.
(the NET has a lengthy footnote explaining the ambiguity with references).
In many ways, it's misleading to single out Piper and Wright in the naming of this stream. There is a long-running debate with many scholars on either side.
But because of Wright's popular works, Piper came out with a popular work critical of Wright to which Wright wrote another book in response.
Moo claims that Wright has made the background of the New Testament the foreground and the foreground the background.
He means this negatively, but there is a positive aspect to this (at least the first part). Namely that Wright draws to our attention aspects of Paul's thinking that have largely been downplayed or neglected. I think that is a large part of Wright's appeal for me.
To use a musical analogy, like Wright is so fond of doing: imagine a sonata whose first and final movements are well known but whose slow movement is rarely listened to. By my attention being pointed to the slow movement, I not only develop a new appreciation of that movement but for the entire piece as a whole and I see a unity I was otherwise blind to.
Wonderful video of a conversation with Tom Wright and Richard Hays, moderated by Michael Gorman:
Hays, in some ways, is even more New Perspective than Wright here.
I could watch Hays and Wright in conversation over the text for hours :-)