I agree with Martin's answer - but there were other additional and compelling reasons.

A rough path (as in the original papers) was a path in the tensor algebra over V with appropriate "p-variation". Without any further conditions one can solve differential equations driven by it; however, the solutions are canonically paths in in operators on functions on the manifold (think the enveloping algebra of the Lie algebra generated by the vector fields) (think randomly evolving heat kernels on the manifold) and are not evolving points on the manifold. Rough paths drive differential equations providing the vector fields come from a Lie algebra structure that is inherited from an associative structure (when truncated to order p etc.).

You can push the solution back down onto the manifold if you have a connection or other strong geometric structures on the manifold. The Ito integral you refer to is such an example. Solutions to differential equations driven by geometric rough paths stay in the integral surface or manifold defined by the original vector fields.

Another reason for the notion is that continuous bounded variation paths are p'-rough path dense in the geometric (and weakly geometric) paths. They are the closure of the classical paths. As all the basic results of integration, differential equations, .. are continuous in these metrics, and identities such as the change of variables identity therefore hold on closed sets of paths, they automatically hold for geometric rough paths if they hold for smooth paths. Paths for which one needs correction terms are more exotic and it is helpful to have language to distinguish them. Perhaps with hind sight, weakly geometric rough paths should be rough paths and rough paths should be diffusive rough paths? But it is too late now.