Errors like this point to the problem with the plugin installation process when additional database tables need to be created. If that has not happened, it points to the problem with database access or something else related to the database server, but it is might be simply WordPress related issues preventing the normal update.
To check this out, open bbPress Toolbox menu -> Tools -> Recheck and Update. This panel will recheck and recreate tables in possible. If not, you will see the error messages. If that happens, you need to create tables manually (via phpMyAdmin, for instance). Here is the SQL schema for all 3 database tables plugin needs: Database Schema.