Hi, Ryan,
I have had another think about this and wonder if your ancestor was in the Waterloo rd Workhouse, long since demolished but was near Bonnar Rd That is also in the Ecclesiastical parish of St John which I had not realised. the residents are listed clearly in the 1881 census. there are several references and other family history references on the Web.
The records for the Bethnal Green hospital are all held at the archives of the Royal London Hospital in Whitechapel
You ask about the diference between a hospital and an infirmary. the short answer is not a lot! Some places notably Bristol still refer to their hospital as an Infirmary as in Bristol Royal Infirmary.
All work houses segregated their residents into men , women and children, even breaking families up. They were then devided again; the able bodied who could work, the idle who would not work, and the infirm who could not work. Each resident was allocated a category on admission by the Medical Officer. The infirm were therefore cared for in the infirmary. In a way it was a bit like a sick bay in a boarding school or other institution. It housed the elderly infirm as well as the mentally ill and physically ill with conditions like TB and bronchitis
The modern term hospital is I think well understood!
Which census did you find your relative in ? It may be possible to use the refernce to find the location of the workhouse
I have a variety of search subscriptions so can try a look up if you PM me the details
Best wishes
dochines