Do Newton's laws of motion imply no physical difference between different inertial frames of reference? I'm a mathematician learning physics from scratch, for my own curiosity and interest. Starting from the basics, I'm trying to get a deep grasp of Newton's laws of motion.
V.I. Arnold describes Galileo's principle of relativity as following (the bold font is my addition):

There exist coordinate systems (called inertial) possessing the
following two properties:

*

*All the laws of nature at all moments of time are the same in all inertial coordinate systems.

*All coordinate systems in uniform rectilinear motion with respect to an inertial one are themselves inertial.


From Newton's laws, we can deduce the existence of inertial systems (the first law) satisfying the second property, and also that the behavior of all mechanical systems is the same in all inertial coordinate systems. If you are in a sealed train, measuring forces between bodies and their resulting motions will not differentiate a train at rest from a train at a constant velocity. But Galileo's principle uses an seemingly "stronger" formulation, stating that no physical phenomenon ("law of nature") will differentiate between the cases.
Are the two equivalent? Does every physical phenomenon stem from the behavior of mechanical systems? Or could we imagine a universe in which Newton's laws are correct but there's still an elusive, tricky experiment to refute Galileo's principle?
 A: 
" Galileo's principle uses an seemingly "stronger" formulation,
stating that no physical phenomenon ("law of nature") will
differentiate between the cases. Are the two equivalent? "

No, the 2 are not equivalent.
There are other laws of nature apart from the laws of mechanics, chief among them are the laws of electromagnetism.
So, the statement that "all laws of nature are the same in all inertial frames" is  a stronger claim than " all laws of mechanics are the same in all inertial frames. "
During Galileo's time, electromagnetism was not known. So, he probably considered the laws of mechanics as encompassing ALL the laws of nature.
Einstein's relativity made a stronger claim that "ALL laws of nature including the laws of mechanics AND the laws of electromagnetism are the same in all inertial frames .
A: The two principles are not equivalent, of course. When Galilei formulated his principle, he had mostly mechanical systems in his mind. You can find the famous passage in the Dialogo sopra i due massimi sistemi del mondo, or you can just read the translation on Wikipedia: Galileo's ship. He talks about jumping, trowing balls, leaking water, swimming fish and flying flies, and argues that all these phenomena happen in the same way when moving at a constant speed inside the deck of a ship.
We now know that the principle can be extended (and slightly modified) to include all (known) laws of nature. And this is best expressed with Einstein's strong equivalence principle:

The outcome of any local experiment in a freely  falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

To address the last part of the question, yes, we could imagine a universe where Newton's laws are correct but other laws are different in different reference systems.
Imagine a universe where relativity is not the correct theory. Newton's theory is. Maxwell's equations of electromagnetism are the same as here. Since they explicitly contain the speed of light, and speed depends on the reference system, the laws of electromagnetism would depend on the reference system.
In particular, this would mean that there is a special inertial reference system in which light propagates at $c$ and Maxwell's equations hold. Every other reference system would have different laws for electromagnetism, that would depend on the velocity of the reference system with respect to the special one.
In this universe, you could always tell if you are moving. In the same way a sailor can determine the velocity of the ship with respect to the water, by looking at the waves it produces. The waves will propagate faster in the direction opposite to the direction of motion. In this hypothetical universe, you could measure the speed of light in different directions and find out your velocity with respect to the special reference system.
