The primary difference is that in the categorical setting one has morphisms that also need indexing.
[1] The actual objects and morphisms in J are largely irrelevant; only the way in which they are interrelated matters.
The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J.
A morphism of diagrams of type J in a category C is a natural transformation between functors.
[2] A cone can be thought of as a natural transformation from the diagonal functor to some arbitrary diagram.
The commutativity corresponds to the uniqueness of a map between two objects in a poset category.