From the beginning, I designed the avl_array with the clear intention of hiding the tree nature of the implementation to the user's eyes. In this concrete case, I can't think of any benefit coming out of exposing the tree hierarchy. In part because of the banlance operations.
That would be one big difference, which has been brought up before in the same context. I don't see a harm to have access to the tree. If you really want to limit such access you can make a separate adaptor.
Would there be any benefit to accessing the data structure as a tree in client code? If so, shouldn't std::map do the same? My opinion is that this is a sequence container that gives certain complexity guarantees, and over-specifying the public interface restricts potential future implementations. The fact that the sequence uses a tree internally is an implementation detail. And actually, given that, I think it should be renamed to something that reflects the strengths of the container, rather than after the implementation structure. (Not sure what a good name would be, but something to think about). Just my $0.02.