This is explained in detail in the documentation chapter Elements —> Structural Elements —> Beam Elements —> Choosing a Beam Element. Of course, it’s based on those two well-known beam theories so literature for mechanics of materials will also help.
But in short words, Euler-Bernoulli’s formulation doesn’t account for transverse shear deformation (cross-section remains normal to the beam’s axis after deformation) and is meant for slender beams while Timoshenko’s formulation takes this effect into account (cross-section may not be normal to the beam’s axis after deformation) and is meant for both thick and slender beams.