This one is a challenge, and one that requires the use of historical lexicons, rather than just the concordance. Most people who use only English translations assume that these are two very different ideas. However, they are actually translations of the same word! The choice by the translators is completely arbitrary, and has created serious misunderstandings among people who are sincerely concerned with faithfulness.
Listing of classical uses of dikaios, (the adjective form), begins with Homer, who is thought to have composed his famous epics, the Iliad and the Odyssey around the tenth century BC. The first appearance of dikaios refers to well-ordered, civilized behavior; being observant of one’s duty to both gods and men. Later, it referred to equality, fairness, and impartiality; to legal precision; to doing or receiving what is right and just. (This was the original English meaning of “righteous”, as well.) It referred to the mutual obligations in contracts.
The noun form, dikaiosune, was also a legal term: the business of a judge in a courtroom. It referred to legally claimed rights or demands; to the vindication of the innocent and the sentence passed upon the guilty.
The verb form, dikaioo, could refer either to the demanding or maintaining of one’s rights, or to being “set right”, to be caused to act properly, or to be treated justly. Therefore, whether vindication or punishment was in view, would depend upon the perspective — or the behavior — of the individual concerned. And the resultant behavior would depend upon the “justice” of the situation — and the judge!
None of these ideas is foreign to the Old Testament prophets, who frequently called for justice among God’s people. Neither is it foreign to the New Testament writers. How the simple concept of “justice” morphed into the much more elusive, theological construct of “righteousness” is unclear. I’m inclined to guess that it happened contemporaneously with the transformation of the Law, (which was designed to promote/create a just and fair society,) into an increasingly complex system of ceremonial and ecclesiastical minutiae that required constant professional analysis, refinement and interpretation!
When people in positions of leadership begin to acquire power over their fellows — be that power physical, political, or theological — justice among the general populace is one of the early casualties. The stern requirement of “righteousness” , defined (and therefore understood) exclusively by those in power, then became a useful tool in retaining and increasing their dominion.
Turning to the concordance, (the back of the book this time — see the introduction to Word Study), it is interesting that the adjective dikaios is translated “just” 33 times, “right” 5 times, and “righteous” 41 times. The context will provide you with some clues to the reasons why different words were chosen. But remember, the original word did not change. The changes were an artifact of the ideas of the translators, not the first writers.
The verb form was rendered “justify” almost exclusively — a word which today is understood far more theoretically than the more accurate “to make just” or “to do justice” would allow. For the noun form, most translators do not touch the idea of “justice” at all, moving exclusively to the term “righteousness.” This can only have been a theological decision, as there is no linguistic reason for the departure. Read an assortment of these references, substituting “justice” for “righteousness,” remembering that it is the same original word in each instance. How would that affect your understanding of any of these passages?
I suggest that the biggest difference would be the substitution of the concept of transformation for the sleight-of-hand idea to which many of us have become accustomed. You know the drill — the statement that “God doesn’t see” us as we are, but “sees” instead the moral perfection of Jesus. Could the originators and the perpetrators of that line really think that the Creator of the universe is so easily deceived? Or would choose to be? If one understands the active nature of the verb dikaioo, it becomes clear that there is no deception or pretense involved. Dikaioo implies a transformation of the lives of those who come to Christ — a new creation — that endows people with the true justice that the Lord Jesus embodies.
A proper linguistic understanding of dikaios, dikaioo, and dikaiosune eliminates another “favorite debate” of folks who enjoy focusing on theoretical theology rather than the practical principles of faithful living: the endless argument over whether “righteousness” is “imparted” (instantly created) or “imputed” (attributed or assumed) — both without regard to any empirical evidence — to an individual who makes a profession of “faith” (see Word Study #1 for this one). Neither of these words appears in the New Testament — nor do most of the terms that refer to theoretical speculation. More to the point of the transformation of life described by all the New Testament writers, would be the recognition that the justice of God, created and exemplified by the Lord Jesus, is carefully implanted in those who are his. It is all a gracious gift, to be sure: but one that, like everything else about such a wonderful new life, needs to be watered, nurtured, and cultivated, in order to grow to reflect with integrity the perfect justice of our Lord, the Giver of Life.
The Pioneers' New Testament 
You’re in good company on this one, Mom, at least in the crowd I’m running with. N.T. Wright has written in his work on “Justification” that it is also primarily a “law-court term.” Here’s a summary of Wright’s position on justification, which has gotten him thoroughly crucified in many Evangelical circles due to his statements that obedient living is part of the picture. A couple excerpts:
God vindicates in the present, in advance of the last day, all those who believe in Jesus as Messiah and Lord (Rom. 3.21-31; 4.13-25; 10.9-13). The lawcourt language indicates what is meant. ‘Justification’ itself is not God’s act of changing the heart or character of the person; that is what Paul means by the ‘call’, which comes through the word and the Spirit. ‘Justification’ has a specific, and narrower, reference: it is God’s declaration that the person is now in the right, which confers on them the status ‘righteous’. (We may note that, since ‘righteous’ here, within the lawcourt metaphor, refers to ‘status’, not ‘character’, we correctly say that God’s declaration makes the person ‘righteous’, i.e. in good standing.)
And
This is why lawcourt imagery is appropriate: the covenant was there, from Genesis onwards, so that through it God could deal with sin and death, could (in other words) put his creation to rights…
This double declaration will take the form of an event. All God’s people will receive resurrection bodies, to share the promised inheritance, the renewed creation (Rom. 8). This event, which from one point of view is their ‘justification’, is therefore from another their ‘salvation’: their rescue from the corruption of death, which for Paul is the result of sin. The final resurrection is the ultimate rescue which God promised from the beginning (Rom. 4).
And perhaps most interestingly, hearkening back to your previous posts on faith:
The ‘faith’ in question is faith in ‘the God who raised Jesus from the dead’. It comes about through the announcement of God’s word, the gospel, which works powerfully in the hearts of hearers, ‘calling’ them to believe, or indeed (as Paul often puts it) to ‘obey’ the gospel…
Interesting. We agree on the legal implications of the word, but not, in this part quoted at least, on the total etymology, which I maintain is much more of an active word. If the legal “justification” has indeed taken place, then it will SHOW in the behavior (read, “resurrection life”) manifested by the recipient. Mr. Wright needs to look at the verb tenses, and see how many of them are PRESENT, not future.
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