Is there any conceptual difference between topological insulator (TI), and topological band insulator (TBI)? And if there is, what would it be?
Topological insulators are materials which are insulating in the bulk, but have conducting states on their surface. These can be realized even in strongly correlated systems where the band theory of solids does not hold.
Topological band insulators form a subset of topological insulators which can be described using band theory.