Analysis Services
is the name of the entire product line for OLAP from SQL Server. The term is often used as a moniker for the SQL Server Analysis Services server, ie. the back end service part of the OLAP line. BI Studio
is the client administration and development tool for OLAP.
This is similar how the relational engine splits into the SQL Server engine (sqlsrv.exe) at the back end and the SSMS tool, the client to administer it.
Both SSMS and BI Studio are based on the Visual Studio, but SSMS is integrated at a very low level and thus appears as a stand alone product, while BI Studio is integrated at a higher level where it appears as a ordinary VS add-on package. Both products (SSMS and BI Studio) are available only through SQL Server and are deployed as part of the 'SQL Server Administration tools' option during setup (if I remember the name correctly). None of them is a part of the Visual Studio instalation, ie. there is no option you can select during VS setup to get the BI Studio. But due to the way the integration works if you do install Visual Studio as a standalone product and BI Studio on the same machine, the BI components will appear integrated into VS: BI projects will e available as 'new project type' selection, along with C#, VB and other project types. As I said, this is because the way BI Studio is integrated with VS.