No, it is not; even though the names are similar they are two different companies.
Bank of Scotland (BOS):
Was founded in 1695 and is now a part of Lloyds/HBOS group
BOS is primarily a mortgage and loan- provider
Royal Bank of Scotland (RBS):
Was founded in 1727 and is part of The Royal Bank of Scotland Group. Currently, the British government has a major stake in RBS. The Group's brands include amongst many others RBS Coutts, NatWest, Citizens and Charter One.
RBS is an international bank that offers financial services to corporates, financial institutions, foundations and public sector in the Netherlands. In addition, RBS provides investment products for private investors. In the USA, UK and Ireland RBS offers financial services to wholesale as well as retail clients.