Delhi is the National Capital Territory of India. New Delhi is the Capital City of India.
Delhi of today is a state that comprises of seven ancient cities, all zipping through the ravages of time. Among them, New Delhi is one.
New Delhi was designed as the capital of the British Empire in India. In 1911, the British shifted their seat from Calcutta to Delhi. Thus, started the construction of an entirely new city, aimed to rival the then beautifully planned ancient Mughal Capital of now Old Delhi or the Walled City.
Modern Delhi = New Delhi + Old Delhi + Outer Delhi.
New Delhi is the Capital seat which houses the ministerial buildings and plush bungalows and hotels. It is beautifully planned and has 47% tree cover, making it one of the greenest capitals across the world! Most of this city was designed by the British architect Edwin Lutyens, especially the Central Delhi region. Today's New Delhi is an extended city, expanding beyond the British Capital to house large areas of South Delhi, few areas of West Delhi and a bit of North Delhi, too. As such, today's New Delhi is the most diverse of the 3 Delhis, having the very green, royal British Capital, green, plush and historic South Delhi, and malls and designs of West Delhi.
Old Delhi is the smallest of all, enclosing only the Mughal Capital, including areas around Chandni Chowk, Red Fort, Jama Masjid, etc and some surrounding areas such as Shahdara, etc. It is restricted to East Delhi only, on either side of the Holy river Yamuna.
Outer Delhi is the largest of them, being the posterior regions of Delhi. It has the most variable form of development, ranging from the highly planned arear like Mayur Vihar, Pitampura and Rohini to the messy Badarpur, Narela, Najafgarh, most of which are Rural and Industrial areas. It covers almost the entire North Delhi, few parts of South Delhi and considerable amounts of West and East Delhis too.
Infact, Outer Delhi and Old Delhi are commonly referred to as Delhi only.
Thus, the overall distinction converges to Modern Delhi = New Delhi + Delhi