What's the difference between classical rigidity and Born-rigidity? In classical mechanics, you have the concept of a rigid body. This notion is incompatible with the theory of special relativity.
In 1909, Max Born introduced the concept of Born-rigidity. He did this to dó make it compatible with the theory of special relativity so before the advent of the general theory. With the introduction of accelerations in the special theory, which lead to the general theory, a paradox came into existence: the Ehrenfest paradox, which led to certain restrictions for the Born rigidity. I don't want to delve deeper into the problems of Born rigidity in general relativity (I just added this for some extra information) because my main question is: what's the difference between classical rigidity and Born-rigidity?
 A: (I am not an expert, any correction is welcomed)
A rigid body is an object that does not deform when applying a force on, it only translate as a bulk. A cube for example can be rotated, moved or flipped to one side or another. But, you cannot stretch it, shrink it or twist it.
A rigid body is an object in which the distance between any two points on it is fixed.
A Born rigid body follows basically the same rule, but with a spin off. Here, you consider a co-moving inertial frame: an inertial frame moving with the same velocity as one of the points on the object; say the center of the cube. In this frame, just like the normal definition of a rigid body, the distance between any two points on the object must be the same.
With that being said, Wikipedia (Date: August 8, 2019) have a beautiful compact definition that goes as follows:
Born rigidity is satisfied if the orthogonal spacetime distance between infinitesimally separated curves or worldlines is constant, or equivalently, if the length of the rigid body in momentary co-moving inertial frames measured by standard measuring rods (i.e. the proper length) is constant

Bonus example:


*

*If you take a cube stationary in $S$ frame, applied a Lorentz transformation on it $\Lambda$, the object will look contracted, giving the notion that the object is not rigid. But it is (at least Born rigid), the distance between any two points on the cube is the same. Measured in a co-moving frame of one of the points on the cube, with its standard measuring rod.

*(Ehrenfest Paradox) If you take a disk and applied a force on the edge, the disk will spin. According to special relativity, the circumference of the disk must contract, but not its radius! How is that even possible? In fact, this paradox is one of the motivations of the general theory of relativity. To keep it short, the disk when the force is applied is not Born rigid, there will be stresses on the material when accelerated. Further, the space of the co-moving frame to one of the points, say the center of the disk, is in fact non-Eucleadian. 
A: In Galilean relativity, we can define rigidity as the requirement that for any two points on an object, A and B, the distance between A and B stays constant.
In special relativity, this definition needs to be modified, because the distance between two points depends on the state of motion of the observer, and different parts of the object may be in different states of motion. So in Born-rigidity, we require that for points A and B that are infinitesimally far from each other, the distance between A and B stays constant in the frame of those points.
