Can someone explain in layman's terms or at least give a summary of the key differences of approaches between Woodin and Sy Friedman regarding set-theoretic truths?
After this great debate, are there any progress being made?
It is only a half-joke to say that for a layman, there is no difference between their approaches. Take the following question for example: Does there exist an uncountable subset of $\mathbb R$ that cannot be put into bijection with $\mathbb R$? Both Friedman and Woodin would say that this is a question that might admit a definite yes-or-no answer, but that the jury is still out and we need to do a whole lot more technical set-theoretic work before we will be in a position to say whether it does admit a definite answer, and if so, whether the answer is yes or no. This already sets them aside from many others, e.g.,
formalists who say that the question does not admit a definite yes-or-no answer, or
platonists who say that the question has a definite answer but who also say that we already know that we will never know the answer.
To even state the differences between their approaches requires considerable technical machinery. Friedman's approach is called the "Hyperuniverse Program." Roughly speaking, his "hyperuniverse" is the space of all countable transitive models of ZFC, and the idea is that these universes are not all on an equal footing; e.g., some models are in a sense "maximal" and this clues us to which set-theoretic statements are true. Woodin on the other hand may be thought of as taking the inner model program as a starting point and arguing that the mathematical evidence points towards something called $\Omega$-logic as a natural foundation for set-theoretic truth. I do not think that too many further details can really be accurately described in "layman's terms" because the technical prerequisites are considerable.
As for whether there has been "progress," both approaches lead to a plethora of technical mathematical questions, and although I do not follow this area, I expect that progress has been made in the same way that any subfield of mathematics makes progress. But if that's what you mean by progress then again I am not sure that it is possible to summarize it "for a layman." On the other hand, a layman could perhaps follow some of the philosophical debates about truth, but then it is less clear (at least for most mathematicians) what "progress" means in the philosophical realm.