Difference

Hello, to all views

I would like to know the difference between “look forward to hearing from you” and “looking forward to hearing from you”.
Best regards,
Robert

They both mean the same and are correct gramatically, though as a native speaker I would say that the second statement possibly suggests that one is hoping for a more immediate reply.

“I look forward to hearing from you” is more formal than “I am (or I’m) looking forward to hearing from you.” You would use the former in a business letter or in an email to an interviewer, and the latter to someone that you have a more casual relationship with.