Tiger Barbs, Green Tiger Barbs, Albino Tiger Barbs, Red Tiger Barbs, Moss Tiger Barbs, <insert here> Tiger Barbs, (for good measure - they are also known in some older books as "Sumatran Barbs"), are all the same fish Barbus tetrazona. They have the same outlook on life, i.e. swim frantically around the tank at Warp 6, chasing each other, or if there aren't others, well, anything else then.
Barbus semifasciolatus, the true Green Barb, (here we go with common names again!), also called the HalfBanded or HalfStriped Barb, is a larger, (3-4" typically can grow bigger if well cared for), peaceful fish which minds it's own business and cruises sedately around the lower half of the tank. They are ideal community fish with things to big to swallow, but are not happy in situations where they are being chased around by more boistrous tank mates.
If you want something to go with a pre-existing squadron of tetrazona, something like conchonius, (Rosy Barb), or nigrofasciatus, (Black Barb, Ruby Barb, Black Ruby Barb, Purple Headed Barb), are easily available. Don't get fooled into buying the "long finned" varieties though, the tetrazona will have them off in short order.
There are a great number of very nice 2-3 inch barbs, sadly, many are never seen in the average dealers tanks.