Both process are used to clean the workpiece surface and remove dust.
Sand Blasting is an abrasive process used to clean surfaces of rust and debris. Typically it is used prior to a painting or coating process. It uses sharp edged media, or sand, to chisel away at the surface a bit like a chipping hammer, but with millions of particles.
Shot Blasting is a conditioning process used to create a layer of compressive stress in the material of the part. This compressive layer, in simple terms, means that you need a higher level of tensile stress to get the surface material to fail. Shot Blasting uses rounded or spherical media of different materials, typically harder than the target material. The quality of the media is continually checked to ensure that the number of broken 'shot' is not too high. It is also usual to have standard test part exposed to the same process as the working part. This is called an almen strip, and by the curve it takes up, indicates the level of compressive stress in the working part surface.