If 1 person says a cow is a 4 legged mammal and another person says a cow is a 4 legged mammal from the family bovidae, is the first person wrong and the second person correct? Of course not the second description is a little more complete. Now if i transfer this to mechanics we can explore what Einstein saw that Newton didn't. Classical mechanics (often, and perhaps unfairly, called Newtonian Mechanics) uses Gallilean transforms to transfrom between inertial reference frames not Lorentz transforms. It is also based on Newtons 3 laws of motion. These laws are (copied shamelessly from wikipedia)
First law
There exists a set of inertial reference frames relative to which all particles with no net force acting on them will move without change in their velocity. Newton's first law is often referred to as the law of inertia.
Second law
Observed from an inertial reference frame, the net force on a particle is equal to the time rate of change of its linear momentum: F = d(mv)/dt. Since by definition the mass of a particle is constant, this law is often stated as, "Force equals mass times acceleration (F = ma): the net force on an object is equal to the mass of the object multiplied by its acceleration."
Third law
Whenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line. Newton's third law is sometimes referred to as the action-reaction law.
We see that the second law is a definition, not really right or wrong, the third law follows largely from the second law. The first law just says if you don't do anything nothing will change. None of these points are wrong as such, the problem lies more with Galilean relativity. Einstein added a further two postulates to mechanics. Again from wikipedia,
* The Principle of Relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.
* The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.". That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.
another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line. Newton's third law is sometimes referred to as the action-reaction law.
The second one is the key one. It is this postulate that leads us to replace Galilean transforms with Lorentz transforms. From this change comes all of the special relativity. So then are the two mutually exclusive? No they're not. When describing an object of low mass and low speed both Lorentz transforms and Galilean transforms give the same answer. So make to my rather trivial example with the cows Newton wasn't wrong, he just didn't realise that he was incomplete. Einstein is quite possibly incomplete as well, although the mixture of special relativity and quantum mechanics used to describe electromagnetism is the most successful theory ever.