Why is Einstein's "Equivalence principle" that new when Einstein formulates it? If I understand it correctly (see here for example), the "Equivalence principle" roughly states that "acceleration" has the same effect as "gravity".
But don't we know, since Newton, they are indeed equivalent, at least from the equation m.a = F = m.g?
 A: There are several different notions of 'equivalence principle' in general relativity. The one you describe is sometimes called the weak equivalence principle. This states that the trajectory of a freely falling test body depends only on its initial position and velocity, and is independent of its composition. In the context of Newton's laws, this is the statement that gravitational mass and inertial mass are equivalent, or the statement that uniform acceleration is indistinguishable from a uniform gravitational field.
The Einstein equivalence principle extends this idea beyond the motion of test bodies to the nature of all physical laws. Let us define a local inertial frame as one in which test bodies experience no acceleration due to gravity. The Einstein equivalence principle then states that in a local inertial frame, the results of all non-gravitational experiments are indistinguishable from the results of the same experiments performed in an inertial frame in Minkowski spacetime.
This principle is clearly more general than the weak equivalence principle. For instance, it allows us to make statements about the passage of light in a gravitational field. By arguing that light must move in a straight line in a locally inertial frame, we conclude that light must bend in the presence of a gravitational field. This is not something that we could have concluded from Newton's laws!
As a final comment, if one thinks a bit more deeply about the two notions of equivalence principle described above, it occurs to one that the inner workings of any reasonable test body used in the statement of the weak equivalence principle are going to be subject to the laws of electrodynamics, quantum chromodynamics, and so on. One can argue that if these forces behaved differently in local inertial frames to inertial frames in Minkowski spacetime, then the motion of test bodies ought to differ also. These ideas are captured in Schiff's conjecture.
A: Yes, we know that acceleration has the same effect as gravity since Newton, but Newton knew both forces are of very different kind. He had to introduce the gravitational constant G to be able to use the same units for the inertial and for the gravitational mass. G is very, very tiny: 6.67 * 10-11 (!)
Einstein declared every acceleration could be explained by gravitational fields. That seems to be correct in very simple cases, but it is not appropriate for the most cases in life. Try to describe vibration and all other accelerations in a simple diesel car by gravitational fields. That would be nonsense. For any physicist it is recommended to separate both causes.
