The cleanest way into this dispute is to notice that both men are liberals in the broad sense — both take individual rights seriously, both reject utilitarian trade-offs that sacrifice some people for aggregate benefit. The disagreement is narrower and deeper: whether justice can be specified as an end-state pattern, or only as a history.
Rawls asks what principles you would choose if you did not know who you would be. Not your class, talents, race, gender, or even your conception of a good life. His claim is that this ignorance forces impartiality: unable to tilt the rules toward yourself, you would choose rules you could accept from any position. He argues you would secure equal basic liberties first and refuse to trade them for economic gain, then permit inequalities only where positions are genuinely open and the arrangement improves the lot of the worst-off. Behind the veil, you would not gamble on landing at the bottom of a society you designed.
Underneath this sits a moral claim that does most of the work: the distribution of natural talents is arbitrary from a moral point of view. You did not earn your intelligence, your health, or the family that raised you. Since you do not deserve the assets, you do not straightforwardly deserve everything they yield — which opens space for treating their products as partly a common resource.
Nozick attacks the pattern itself. Suppose a distribution you consider perfectly just. Now a gifted basketball player — Nozick's example was Wilt Chamberlain — agrees to play, and a million people each freely pay a small sum to watch him. He ends up far richer than everyone else. The starting point was just by hypothesis, every transfer was voluntary, and nobody was wronged. If the resulting distribution is unjust, then justice requires forbidding transactions between consenting adults, or continuously undoing their results.
That is his general argument: liberty upsets patterns. Any principle specifying how things should be distributed will be violated by free exchange almost immediately, so maintaining it demands either constant interference or forbidding capitalist acts between consenting adults. Justice, he concludes, must be historical — concerning how holdings came about, not what shape they form.
Each faces a serious objection. Against Rawls: the veil argument arguably assumes a particular attitude to risk, and it is not obvious that rational choosers would maximise the minimum rather than gamble; critics also argue he under-weights what people deserve through effort. Against Nozick: his theory includes a principle of rectification for unjust acquisition, and since actual holdings descend from centuries of conquest, slavery and expropriation, his own framework may demand redistribution on a scale that dwarfs anything Rawls proposed — a consequence he acknowledged but never worked out. That neither position has been abandoned, half a century on, suggests both are tracking something real.