A new government wants to take control. They want to give everyone $5 a month. Because they believe that is equality. There are rich people and poor people. So if the government gives $5 to everyone they are making the rich richer and the poor a bit better off. But still the two classes are not equal However that government says they are.
There's justification missing in the scenario. It's unclear why the government takes that measure. Because under some justifications it could follow Rawls' ideas while under others it wouldn't at all. The latter seems much more likely (more on that below).
The SEP describes the guidance that results from Rawls' difference principle like this:
If a system of strict equality maximizes the absolute position of the least advantaged in society, then the Difference Principle advocates strict equality. If it is possible to raise the absolute position of the least advantaged further by having some inequalities of income and wealth, then the Difference Principle prescribes inequality up to that point where the absolute position of the least advantaged can no longer be raised.
For Rawls, the absolute position of the least well off is where to look when deciding on which measures should be taken. It's likely that the action in the scenario doesn't favor the least advantaged enough. A better option could be an amount that is diminishing the better the position of advantage becomes. For example something like: the worst off get 10$, the best off get 0$, in between it scales. But it depends on the results that come from the amount that the government gives out. If and only if the constant amount (the 5$ for everyone) somehow results in economic growth that raises the absolute position of the least advantaged higher then the diminishing amount then it's the better option. Otherwise it's not.
It could also be the case that 5$ simply aren't enough, or that other measures that result in economic growth which raise the absolute position instead of just giving out some amount of money are better.
Simplified, Nozick thinks that the job of distributive justice can only be to rectify things that were aquired or transferred unjustly. How the distribution looks doesn't matter as long as the process was just. He gives guidelines how a correct aquisition or transfer are supposed to look. Whereas Rawls thinks that improvement of that absolute position of the least well off matters and distributive measures should be designed that way.