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.