SAP BW and HANA are part of the database and technology set of products and has nothing to do with SAP BusinessObjects BI. SAP BW is part of the SAP Netweaver stack and is predominantly used as an Enterprise Data Warehouse (dedicated to reporting as opposed to an Operational Data Store).
SAP BI is a product suite that is used for reporting and can be deployed over a Microsoft EDW environment (SQL) or Netweaver (SAP BW) environment. SAP BI includes dashboards, explorer, web intelligence, etc. SAP BI is part of the BO (BusinessObjects suite of products). BO also has other product suites such as Enterprise Performance Management and Enterprise Information Management.
In turn BusinessObjects is part of the SAP Business Analytics set of products which also includes SAP GRC (Governance Risk and Compliance).
This is as far as I know at this point in time, but it used to be different before SAP purchased BusinessObjects. Then BI and BW were pretty much "interchangeable". Also, no guarantee how long this will be true.