One reason why modular motives are of interest is physics oriented, which may not be what you're looking for. In string theory modular forms arise naturally via the propagation of the 1-dimensional string itself, because the 2-dimensional worldsheet that is swept out by the string supports a conformal field theory. On the other hand, the string worldsheet is thought to be embedded in a spacetime manifold. The question then becomes wether the compact dimensions of this spacetime manifold can be constructed by considering the motives associated to the string theoretic modular forms. This strategy has been shown to work in some examples. At least in those cases one can view the modular motives as providing a string theoretic explanation of the extra dimensions, given e.g. by Calabi-Yau varieties.
As regards to the work of Deligne and Scholl, their construction is concerned with motives associated to the Kuga-Sato varieties. In the application just mentioned the focus is more on Calabi-Yau varieties, and the goal is to construct motives arising from such spaces.