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It is a matter of choice for each individual customer. Private sector companies are perceived to much quicker in delivery, but Public Sector companies are fast catching up on this front. Factors like trust, speed of delivery and claims efficiency should be kept in mind while making your choice.
On the product side there isn't too much of a difference between the public sector and private sector insurance companies - in both life and general insurance space. Private insurance companies may have more variants to cater to individual requirements, while public sector insurance companies score very heavily on their reach of distribution network.
Ultimately insurance is something you buy for peace of mind - so buy from a company with which you feel confident!