Normalizing results in a more uniform grain structure (it can result in a finer grain structure) and improved mechanical properties. As a by-product of normalizing, residual stress can be removed.
PWHT is a thermal treatment to accomplish two things; reduce residual stresses from welding and temper the weld heat affected zone. There are two types of PWHT; a reheat treatment (like normalizing, or annealing, or a quench and temper) or a subcritical PWHT. The subcritical PWHT is only performed to avoid altering the original base metal properties.
No, you do NOT normalize the weld heat affected zone from a subcritical PWHT, unless you perform a separate normalizing heat treatment.