Is The Great Plain of North India and The North Indian Plain the same place?

Yes, these are both names for the same place.  This place is also known as the Indo-Gangetic Plain.

This area covers most of the north and east of India as well as much of Pakistan and parts of Bangladesh.

To the north of this area are the Himalaya Mountains, which are the source of the rivers that have laid down the sediments that form the plains.  The most important of these rivers are the Indus and the Ganges.  The North Indian Plain is bounded on the south by the Great Plateau of Peninsular India.