I remember this episode of NOVA; I didn't like it.
Time is the same for everyone - if you mean the time which they experience, and measure with their own "wristwatch"; this is called "proper time" in the lingo of special relativity.
According to Newton, this is the only time there is - the same everywhere, flowing equally in all places and all times.
But by Einstein's day it was known that something was wrong with this description - it was incompatible with Maxwell's equations for electromagnetism, and some disturbing experimental results were indicating that some changes were required. Poincare was the first to elaborate, correctly, on the "relativity of time" - and gave it this name. You can find some of the quotes and references here:
http://en.wikipedia.org/wiki/Relativity_priority_dispute
The answer was that while "proper time" works the same for everybody, if you watch the "proper clock" of something which is moving fast relative to your position - then you will "see" their clock running slow. Furthermore, if multiple observers are in motion relative to each other, they will disagree on which events are simultaneous.
However, Special Relativity tells us how to work with this, and make sense of it all. A good place to start is Taylor & Wheeler's "Spacetime Physics". A few new tools are required, but nothing beyond algebra, and some patience.
When you are done you will see that while time is not an illusion; instead it must be treated as one of four dimensions of spacetime, but following rules that differ from Newton and Galileo ... rules that are compatible with Maxwell's electrodynamics.
BTW, there is an immense amount of experimental support for Special Relativity; it is one of the best tested theories available. Plus if it were wrong, then there would be inconsistencies within Maxwell's equations - and they are in use everywhere for electrical and electronic devices, from motors to cell phones, from mirrors to lasers.