As I know about the roles and profiles. In previous version of SAP (till 4.6) Activity groups were known as role(in 4.7 and above).
A role contain profiles. It can contain more than one profile.
Roles describe the layout of the user menu.
and profiles tells the authorizations for the particular profile(authorizations comprises of authorization objects which further are composed of authorization fields), a user access to different objects in the SAP system.