Zig's got it essentially right, elgarak, sorry to disappoint you. The equivalence principle implies that an accelerated frame of reference is equivalent to a motionless frame of reference in a gravity field, and that there is no local experiment that you can perform that will tell the difference between acceleration and a gravity field.
The actual statement is that you can perform a local experiment in any inertial frame of reference and you can't tell the state of motion of the frame. In other words, it might be floating freely in space, at any speed, or it might be accelerating under the influence of a gravity field, and you can't tell the difference. A frame accelerating under the force of gravity is inertial just as the freely-floating one in space is; it is therefore from this point of view correct to refer to free fall as "zero g," although it is not from other points of view.
In practice, because of the tidal effects that Zig mentions (and another effect he doesn't, hang loose, I'll get there in a minute), you can measure the difference- but it is subtle.
The other difference is that because a gravity field generated by a massive object is spherical, the vector of the forces on objects that are separated angularly with respect to the center of mass of the massive object are different. Each points toward the center of mass, and as a result, if (for example) you were free-falling toward the Moon's surface along with two balls on either side of you, you would see the balls slowly move toward you. This of course would not happen under the influence of a "plane gravity field," but we don't know of any physical way to generate such a field in the real world (short of actually making a massive plane and suspending some objects over it, or allowing them to fall toward it, a major engineering project to say the least).
ETA: Zig would no doubt have me point out (correctly) that both the stretching along the force vector, and the compression orthogonally to it, are tidal effects, although only the first is generally mentioned in discussions of the actual tides created on the Earth by the Moon and Sun.