It's a distributed version control system, implementing a sound mathematical model of software development (patch theory). In this sense pijul is like darcs, but it doesn't suffer from exponential-time edge cases when merging; hence it should be a pareto improvement.
Of course, the elephant in the room is git: when git came out, most darcs users (eventually) switched over to it. Hence there's not many darcs users left for which pijul would be a no-brainer. There are many git users, most of whom have never used a patch-based DVCS, who would need a more compelling reason to switch.
Personally, I'm glad the patch-based approach is still going, as it's always good to have competition. Plus, having a sound underlying model usually brings insights to a problem which aren't at all clear in more 'cobbled together' solutions like git (e.g. commutativity of patches is quite a natural property in darcs/pijul, which doesn't directly make sense in git due to its "commits" specifying their parents).
I haven't made up my own mind, and certainly intend to try this out.
But I want to point out that your first paragraph offers no answer to the question, same as the title here doesn't. And I don't say that to be snarky -- what you list are implementation details, not an answer to the question. As a developer I don't care what kind of theory my program is built on I just want to know how it will save me time and give me new functionality. (In my spare time I can think the math is cool and be content with that).
(Even more specifically -- git is a DVCS already, it's just most people seem to use it with a central server like GitHub, so personally I also didn't understand what the page meant when calling that one of pijul's advantages).
In principle I do care about the fundamental model being correct - that can often bubble up to fewer (or cleaner) edge cases. Without help though, or many hours of usage I can't mentally map that model to what it is actually going to be like to use.
Under what real development scenario(s) would Pijul's approach help me?