Cube can (and arguably should) mean something quite specific - OLAP artifacts presented through an OLAP server such as MS Analysis Services or Oracle (nee Hyperion) Essbase. However, it also gets used much more loosely. OLAP cubes of this sort use cube-aware query tools which use a different API to a standard relational database. Typically OLAP servers maintain their own optimised data structures (known as MOLAP), although they can be implemented as a front-end to a relational data source (known as ROLAP) or in various hybrid modes (known as HOLAP)
I try to be specific and use 'cube' specifically to refer to cubes on OLAP servers such as SSAS.
Business Objects works by querying data through one or more sources (which could be relational databases, OLAP cubes, or flat files) and creating an in-memory data structure called a MicroCube which it uses to support interactive slice-and-dice activities. Analysis Services and MSQuery can make a cube (.cub) file which can be opened by the AS client software or Excel and sliced-and-diced in a similar manner. IIRC Recent versions of Business Objects can also open .cub files.
To be pedantic I think Business Objects sits in a 'semi-structured reporting' space somewhere between a true OLAP system such as ProClarity and ad-hoc reporting tool such as Report Builder, Oracle Discoverer or Brio. Round trips to the Query Panel make it somewhat clunky as a pure stream-of-thought OLAP tool but it does offer a level of interactivity that traditional reports don't. I see the sweet spot of Business Objects as sitting in two places: ad-hoc reporting by staff not necessarily familiar with SQL and provding a scheduled report delivered in an interactive format that allows some drill-down into the data.
'Data Mart' is also a fairly loosely used term and can mean any user-facing data access medium for a data warehouse system. The definition may or may not include the reporting tools and metadata layers, reporting layer tables or other items such as Cubes or other analytic systems.
I tend to think of a data mart as the database from which the reporting is done, particularly if it is a readily definable subsystem of the overall data warehouse architecture. However it is quite reasonable to think of it as the user facing reporting layer, particularly if there are ad-hoc reporting tools such as Business Objects or OLAP systems that allow end-users to get at the data directly.