If in both cases if there is an application of Shear Stress then why do we have Screw dislocation in one case and Edge dislocation on the other?
For edge dislocations, the defect line appears to move parallel to the direction of application of shear. That can happen if there is an extra plane of atoms in the lattice that creates the dislocation core.
In contrast, the defect line in screw dislocations moves perpendicular to the direction of shear. An extra plane of atoms is not necessary for screw dislocations to form. Rather, the process is very similar to ripping a strip of paper in two.
There are numerous resources on the web that discuss the issue, e.g., https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/Material_Deformations_Workshop.pdf and http://www.ndhu.edu.tw/ezfiles/29/1029/img/3034/Chapter4-1_1041020.pdf