Sunderbans fall in the Bengali sphere of influence, they have been known to be a refuge for the smaller (but rather more ferocious, I have read that salt water makes them more violent) Bengal Tiger. From what I know, they are the livelihood for various tribal villages around those areas. Historically, they have always been known to have tigers, and have had a rather ominous reputation, especially amongst North Indian people due to this association.
Ecologically, the forests are significant as they are uniquely situated (right before the Bay of Bengal) while socially, they are probably an important heritage site for the Bengali folks of South Asia.
I hear that they are also currently used agricultural expansion in the East of India, near the coastal regions:
Chandra, Ganesh, Diversification for Sustainable Agriculture in Sundarban, West Bengal (July 11, 2012). Paper presented in National Seminar on 'Coastal Resources and Their Sustainable Management: Issues and Strategies, Bidhan Chandra Krishi Visvidyalaya, Mohanpur, West Bengal, 2005. Available at SSRN: https://ssrn.com/abstract=2103496