Differential forms are used in geometric measure theory to define currents. One use of currents is as a generalization of submanifolds, with better compactness properties. That is, it's easier to show that a subsequence of currents converge to a current. This can give one approach to the minimal surface problem; see the encyclopediaofmath.org article on currents.
Maybe someone else can give some better detail or more uses of currents.
