STAAD.Pro results do not match with your hand calculations because shear deformation is considered during STAAD.Pro analysis, while this is not considered in your hand calculations.
Section 1.7.1 of the Technical Reference Manual states the following:
STAAD automatically considers the additional deflection of members due to pure shear (in addition to deflection due to ordinary bending theory). To ignore the shear deflection, enter a SET SHEAR command before the joint coordinates.
Usually, in ordinary bending theory these deflections due to pure shear are not considered. So, if you want that STAAD.Pro deflection results would match your hand calculations, you need to ignore the calculation of deflection due to pure shear by entering the command SET SHEAR in the input file before the JOINT COORDINATES line. With this command you can exactly match simple textbook beam theory results.
You can read some more information about this in STAAD.Pro Technical Reference Manual chapter 1.7.
The difference between the global and local deflections can be seen in the diagram below:
