A Riemannian metric on the manifold provides a (noncanonical) isomorphism between the cotantgent space and the tangent space. Thus, they have the same smoothness properties. However, many definitions are more natural on the cotangent bundle.
For example, if we have the cotangent bundle, it is easy to define a canonical symplectic form on it, as an exterior derivative of a oneform. The one form assigns to a vector in the tangent bundle to the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold). Proving this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on R^{n}xR^{n}. But there the one form defined is the sum of y_{i}dx_{i}, and the differential is the canonical symplectic form, the sum of dy_{i}dx_{i}.
If the original manifold was the set of possible positions, then the cotangent bundle can be thought of as the set of possible positions and speeds. For example, this is an easy way to describe the (nontrivial) phase space of a three dimensional pendulum: a weighted ball able to move along a sphere. The above sympletic construction, along with an appropriate energy function gives a complete determination of the physics of this system.
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