The name '
Geometric algebra' is preferred when the
geometric
properties of the system are emphasized. This is usually desired in
physics,
engineering and other
applications, because this algebra helps people
visualize the problem and understand it in a more
tangible way.
On the other hand, an algebra as such is completely abstract. All possible calculations of Clifford algebra can be done on an abstract level, for example in a matrix representation. If this is sufficient, it is quite pointless to use geometric algebra in the first place.