There is no absolute justification for physical laws. In fact it is best to drop the term law. With Isaac Newton's De motu corporum, book 1 of the Philosophiæ Naturalis Principia Mathematica, the concept of physical laws emerged. Newton's laws were seen in the 18th and 19th century as close to what Moses brought down from Mount Sinai. This idea carried to thermodynamics, where again there are three laws, and somewhat into electromagnetism. With the 20th century the idea of laws was replaced with theory. A theory is a consistent hypothesis on some order of nature that is supported by measurement and observation.
The advances in relativity and quantum mechanics made it clear that Newton's laws are approximations that hold in some domain of experience that is limited. This domain of experience require, velocities much slower than light and comparatively small masses or masses much larger in radius than the Schwarzschild radius for relativity and systems with action $S~>>~n\hbar$ and much larger than wavelength for quantum physics. Outside of that domain of experience Newtonian mechanics fails, and as such they can't really be regarded as laws. They are called laws more as a matter of tradition than anything else.
The state of affairs is not likely to change in any philosophical sense. It is possible that there is final physical theory, which in some operational sense could be called a set of physical laws. This is quantum mechanics, which is not formulated according as laws, but as physical axioms we call postulates. I say this in light of the fact this could be what builds up spacetime. I wrote a post here on SE on how this can be in some fundamental way with an isomorphism between the Tsirelson bound with a line element . Other approaches involve van Raamdsdonk's idea and Sean Carroll has advanced the idea of a combinatorial network of Hilbert spaces in general entanglements. These hypotheses could solidify into more solid theory in the coming decades, and they might be as close to being a law as physics will ever come.
Maxwell's equations are instances of Yang-Mills equations. These are really best understood quantum mechanically. There is a rich underlying theory and mathematics on this. In particular there are the theorems of Uhlenbeck, Freed, Atiyah and Donaldson on the structure of four dimensional manifolds. In addition the gluon in QCD, which is a color charged gauge boson (two charges) can form a colorless di-gluon, which opposite color charges, so that the triplet entangled state is identical to the graviton. It is the possible that the S-dual of the gluon forms in this way the actual graviton. By these means the Maxwell equations, an abelian $U(1)$ gauge principle, and the other gauge fields are a feature of quantum foundations.
We might then say that quantum mechanics is a sort of founding logic to the physical world. The questions are not over, for we have difficulties understanding what is meant by physical reality in quantum physics. Is quantum physics $\psi$-epistemic, as with Bohr and the Copenhagen interpretation, or is it $\psi$-ontological as with the Everett Many Worlds? It seems the human brain that constructs our intuitive understanding of the world in a macroscopic and classical way has troubles understanding some quantum foundations. Physicists have then had a history of imposing quantum interpretations, all of which involve some extra ingredient.
Finally, physics differs from mathematics in that mathematics is a quest for any possible set of relationships between abstract objects. Mathematics is then in a way almost a logical form of art. Physics is more concerned with addressing very similar questions over and over and finding what set of principles can be appealed to that are supported by new observations. Physics is an empirical science, while mathematics is not so much. However with computers there does seems to be a sort of synthetic empiricism developing in the math world.
As for the relationship between physics and mathematics, that is something we are not likely to every understand. The earliest idea of Plato's idea of ideal and real forms, with language (logos) spanning the two. That gets a bit mystical for most physicists, though some mathematicians hold to something like that to justify objectivity to mathematics. In other world mathematics is not only just a sort of word-symbol game, but does involve things that are "real." Max Tegmark has a sort of final multiverse concept of world based on all possible mathematics, but I have a hard time seeing how that can be supported very well by observations. We will have to see how that develops, and there are people who take Max seriously. We may though find that we can never understand how physics and mathematics are unified in some whole way.