Fortuniana, aka Hybrid Banksia, introduced by Robert Fortune in the UK in 1845 and often labeled as a Lady Banks white
http://www.helpmefind.com/rose...
and White Lady Banks aka Rosa banksiae banksiae, introduced in 1807 (Tall, climbing, thornless)
http://www.helpmefind.com/gard...
are 2 different plants, obviously easily confused by using common names.
Fortuniana is probably one of the healthiest roses around, it has small double white violet scented blooms.The canes are gawky and need support, growing into a straggling, scrambling climber. It is mentioned that it has few prickles, not entirely thornless. It is quite similar to Rosa banksiae except that all it's parts are larger. The flowers are somewhat messy close-up and much larger than other banksians and sometimes display a button eye. It is widely used as rootstock in warmer tropical areas with sandy soils and the southeastern corner of the US, Florida being the most popular.
White Lady Banks was the first Banksian rose to be introduced to the West in 1807 by William Kent, who worked for the director of Kew, Sir Joseph Banks. It has large clusters of very double flowers, also violet scented, and is enormously vigorous and is the one grown in Tombstone.