Not much, I'd say.
For one, it bears no moral significance. You can be a net negative scumbag and understand Principia Mathmatica.
Second, it's a formalization of mathematics. It doesn't necessarily yield relevant new insight to advance homo sapiens beyond it's (quite narrow) scope. Understanding Principia Mathmatica doesn't mean you'll know how to make penicillin, for example.
Third, even for pure mathematical work, it's not that interesting in the end. Tons of great math work is being done that has no relation to it.
https://news.ycombinator.com/item?id=25118372
Several of them are getting pretty far, and none of them are directly using PM notation (although all of them ultimately owe something to PM). You can learn more math, or contribute more to math, nowadays by learning one or more of these systems. To my knowledge, all of them are eager for new volunteers to continue formalizing theorems in their respective databases.
It's important to separate historical importance from importance to understanding by most people.
The presence of Whitehead & Russell's Principia Mathematica (PM) in history is a very important historical event. In at least one list, it's one of the most important books in the 20th century. PM shattered any argument that mathematics couldn't have rigorous foundations. PM instead showed that it was possible to start from relatively simple axioms and really prove all the way up to a higher level.
PM is NOT, however, important to read or understand itself in detail, as it's been superseded by later works. Syntactically, PM uses a dotted notation instead of parentheses that most people find hard to read & generally undesirable. From a deeper point-of-view, PM uses a set of axioms built on a hierarchy of types. This hierarchy of types was an early effort to avoid various paradoxes. Almost no one uses a hierarchy of types like this today; it's far more common to use Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Even if you wanted to use a hierarchy of types, which is rare, you're probably more likely to use Quine's "New Foundations" which provides a much simpler foundation for mathematics.
It's not important to understand PM in detail, but knowing that it's possible has inspired many more recent efforts, and that is more important.
I argue in my video "Metamath Proof Explorer: A Modern Principia Mathematica" that the Metamath Proof Explorer (MPE) is a successor to PM. MPE uses modern notation (e.g., parentheses) and ZFC set theory combined with classical logic, so it uses modern conventions, but it also tries to prove "everything" from a very few axioms. Thankfully, we can now use computers to verify every last step (with multiple independent implementations), so we can have much more confidence in the proofs. Here's my video, I think you might like it: https://www.youtube.com/watch?v=8WH4Rd4UKGE
That video focuses on MPE, but of course there are several efforts inspired by PM. Understanding the impact that PM had on mathematics is, to me, the important thing.
However, we still have
* mathematics
* formalized mathematics
* mathematical research
despite the incompleteness phenomenon. All of them are still making progress, even often using approaches that are no more powerful than PM's approach. There are still interesting and important conjectures that can be resolved by a PM-like system, but haven't been resolved yet.
I'm sure lots of mathematicians were deeply disappointed when they understood Gödel's result, but it didn't persuade anyone that mathematics was over or that we should no longer work within formal systems!
>* mathematics
>* formalized mathematics
>* mathematical research
Yeah, none of them are based on or rely in any way on PM.