I haven’t done much QM research, so hopefully you guys can easily answer this:
I was told that the closing distance between two cars moving toward each other at 60mph is not 120mph, but rather 119.999999999…mph basically due to the fact that nothing can travel faster than light.
Is this still relevant in the Newtonian world of 60mph cars?
First of all, this has nothing to do with quantum mechanics (which describes the apparently probabilistic behavior of particles and systems of particles on a level roughly defined as sub-atomic), and second, no it is not true, or at least not insofar as the question is stated. The motion of cars moving toward each other is relative motion, and on any normal everyday scale classical mechanics gives results that are useful out to any practical degree of measurement.
If you want to invoke special relativity then it is true that, for an observer that has been accelerated to motion that a clock firmly mounted in his reference frame will appear to go more slowly than one in an inertial reference frame (and thus, an object coming at him will appear to be moving slightly slower relative to him than it would appear outside the frame), but two cars measured to be going 60 mph directly at each other from the objective frame of a bystander are moving at a resultant relative velocity of 120 mph. As a practical matter, it would not be possible to measure the difference even from within the car. The Lorentz factor would be 1/sqrt(1-0.8e-15), which is too many decimal places for my HP scientific calculator to even solve.
Read the “Clock delays and rod contractions: more on Lorentz transformations” section and the couple below in the Wikipedia article cited above. That should clarify this aspect of special relativity.
But, yes, to address one part of your question; all special relativistic effects are due to the constancy of the speed of light for all observers, no matter their relative motion.