HI, Production engineering is the study of manufacturing technology with management science whereas mechanical engineering is concerned with mechanical systems, thermodynamics, kinematics and energy and related fields.
The difference lies in Mechanical engineers focus more on product development works to design and manufacture new products for customers whereas Production engineers focuses more on making manufacture more resourceful. Product engineers work on the products and process designed by mechanical engineers. SO this is the only major difference.
Production Engineering is one of the roles of Mechanical engineers.

They both can be  inter-related.
Hope this helps.