Since West Bengal is now officially called 'Bangla' I will call it Bangla.
80% of Bangla people have Bangladesh roots. Bangla people fled Bangladesh to India due to religious persecution of radical Jamaatis / fundamentalists. These people are called '
Bangal' (In Hindi it means land of Bengal but in Bengali the word is used to refer people of Bangladesh land, so two different meaning).
@Rain Man 's family was one of such unfortunate family who got kicked of from Bangladesh at the time of partition. Its really sad. So they are always uncomfortable talking about the past. India sheltered Bangal people in Bangla and they are very thankful to India for this reason. Its our fault that we didn't keep our own people in our country but it doesn't mean they are not Bangladeshi.
Bangladesh is not a new name. The whole Bengal was one time called Bangladesh. Ask your Bengali friends if I am wrong. Still people of Bangla (West Bengalis) call 'Banglar Mati, Banglar Jol' (Bangla's soil, Bangla's water), not Bharoter Mati, Bharoter Jol.
There is a popular proverb in Bangla. 'What Bangla thinks today India thinks tomorrow.' Here they themselves make a distinction between Bangla and India.
When I say Bangla is de facto Bangladesh I am not wrong or exaggerating.