Too much to name, most importantly he makes explicit much of what often remains implicit in all areas of science and gives the clearest guide I've ever seen forward. The clarity begins with which Poincaré describes the psychological process of basic mathematical reasoning, laying bare clear differences between arithmetic reasoning and logical inference. From his exposition it is evident how this process shapes what mathematics is, from foundational questions, to basic definitions, to mathematician philosophies, to entire research programmes, down to whether one comes to find interest in these matters at all.What do you like about it?
This exposition on mathematical thinking is threefold, namely by classifying his findings using Kantian terminology, he not only simply demarcates different schools of thought within mathematics, but also makes explicit the limitations which opposing viewpoints bring into mathematical theory with them and why, while at the same time making verifiable/falsifiable hypotheses about the actual psychology and sociology of mathematicians as well, explaining the naturally occuring differences in the choices of approaching and grasping subjects by different people.
The parts on physics are historically especially interesting to read as they are the thoughts of probably the greatest living physicist at the time of the cusp just as classical physics is becoming modern physics. Just reading only the first part of this book makes clear that Poincaré, truly was the last universalist, incorporating at the highest level mathematics, physics and philosophy in such a way not seen anymore anywhere since, especially not in todays age of specialisation. It is also very interesting to note that Feynman's Messenger Lectures on The Character of Physical Law pretty much seem to be to a large extent a dumbed down summary and extension of Poincaré's book.
I believe very much can be gained, not simply for mathematicians and scientists, but for any school child going into any direction, if they could step out of their time and join Poincaré to see all the popular schools of thinking while they were being developed and so then choose themselves instead of just getting a particular view rammed down the throat as is conventional. For example, I think an actual educational system, taught by pedagogically gifted teachers, based on Poincaré's book is capable of producing an entirely new generation of groundbreakingly novel interdisciplinary thinkers.
The hope is of course that this might exacerbate knowledge akin to the modern naturalistic view of network science in comparison with the classical purist view of graph theory, but then for all disciplines. Such a transition would be capable of enabling today's and tomorrow's generations of carrying on successfully into a world where automatisation is increasingly chipping away at tasks requiring human ingenuity without necessarily leaving anything interesting to do behind in its place.