Question

the differences and similarities between the planned cities built during the British period and those built after independence

Answer 1

Don't forget that the British Empire was once described as the 'Empire on which the sun never sets'. Maybe you're thinking of New Delhi, the extension to Delhi after 1937, when the Indian capital was moved from Calcutta (Kolkata) in 1911.
24 state capitals in India have 'planned' cities, or extensions of existing cities.