The Barua Buddhists of Bengal are often said to be indigenous and continuous practitioners of the Buddhist religion among ethnic Bengalis. That is, they descend from the Buddhist communities of Bengali that flourished during the Pala period, and went into decline during the Muslim period, to disappear on the whole. The claim here is to indigenous status.
The Rohingya, in contrast, often make assertions that they are deeply rooted in Arakan. And, they disavow identification as Bengalis. Their language is clearly closely related to that of Chittagong, and it is not usually written in the Bengali alphabet.
Though I am open to being disproven, over the years in my research on the “Barua”, it seems that in the vast majority of cases these “Bengali Buddhists” descend from Tibeto-Burman people who adopted the Bengali language (or a Bengali-related dialect) and settled in and around Bengalis. They are concentrated in the far Southeast of Bangladesh, and often the boundary between themselves as the Theravada Buddhist Chakma, who retain tribal identity but now mostly speak Bengali, is fluid. The Barua are now Theravada Buddhists, which is the tradition of Sri Lanka and Southeast Asia, and not ancient Bengal.
So basically what I’m saying is this: Buddhist tribal people from the east have assimilated into a Bengali identity, and claimed indigeneity through asserting affiliation with the small Barua ethno-religious group. Meanwhile, in Arakan, peasants who migrated over the last few hundred years from southeast Bengal, have rejected assimilation into the Bengali identity, unified around the standard high culture dialect, and created something distinct.