The initial original problem with the indices was that they were used to label coordinates, so mathematicians preferred more and more coordinate independent operators, while physicists preferred continued to use indices. Then, Penrose realized that it has to be something beyond the indices that makes them useful - mainly the Einstein summation convention - and proposed the abstract index notation. This notation is almost identical in form with that of coordinate indices, but it is invariant, like the notation used by mathematicians, and maintains the simplifications due to the use of indices. The indices are not interpreted as labeling coordinates, but as representing the type of vectors and tensors and how they act on each other.
I think that there are advantages and disadvantages in both notations. Though, many tensor operations, especially contraction and type change, are easier to define and perform by using indices.
The following fields can benefit of this notation: Linear Algebra, Representation Theory, Group Theory, Differential Geometry.
This notation can naturally be related to Penrose's diagrammatic notation.