For the contact algorithm - the difference is how to compute the penetration. For a set of simple geometries - sphere, cylinder, plane - penetration is computed even in the closed form. For more complicated NURBS is anyway simplified. Discrete rigid would mean -- application of conventional contact algorithms (Node-To-Surface, Surface-To-Surface) to already meshed (discretized body) and fix all Degree of Freedoms of "discrete rigid" nodes. I have more about the contact kinematics with bodies described analytically in my books (one of which is free down-loadable (Internet License) from my RG account in Contributions/Books )
An analytical rigid does not need to be meshed, and instead can be described by an analytical function. A primitive such as a sphere or a plane would be an example of something that can be an analytical rigid. Discrete rigids must be meshed, and would be something like a CAD surface definition.
As the former respondents mentioned is that the discrete rigids can be used when complex geometry is analysed. I very often delete one of the surfaces of a solid, hence a shell/surface structure is created of the remaining geometry. I mesh the surfaces with a very fine mesh consisting of discrete rigid elements. Resloving the interior is therefore not needed and this most often create a bad surface representation. Subsequnetly I have a very good geoemtrical representation and can easily resolve a good contact condition with the other involved part. This strategy is valid if the part is substantially harder than the others and can be treated as "rigid"!
For the contact algorithm - the difference is how to compute the penetration. For a set of simple geometries - sphere, cylinder, plane - penetration is computed even in the closed form. For more complicated NURBS is anyway simplified. Discrete rigid would mean -- application of conventional contact algorithms (Node-To-Surface, Surface-To-Surface) to already meshed (discretized body) and fix all Degree of Freedoms of "discrete rigid" nodes. I have more about the contact kinematics with bodies described analytically in my books (one of which is free down-loadable (Internet License) from my RG account in Contributions/Books )
People have answered to you very well, but I would like to say that computational time cost of a contact between a mesh and an analytical rigid body is, generally, lower than a contact of a mesh with a discrete rigid body. So, if your simulation of the rigid part may be sketched with an analytical rigid body, I would recommend you this option instead of discrete rigid bodies.
On a very simple note: Analytical rigid surfaces are defined mathematically using lines and arcs, whereas Discrete rigid surfaces are meshed using rigid elements. Basically, if your rigid surface can be created by extruding or revolving lines and arcs then use analytical. If not, then you have to use discrete rigid.