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Though HDF5 was likely the best choice out of the existing data formats, there were some key drawbacks that are important to the use-cases of ASDF. A full discussion on the subject is contained in the ASDF paper, but a short summary follows.
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HDF5 is an entirely binary format, where FITS and ASDF support human readability and edit-ability with very primitive tools.
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HDF5 is not self-documenting. Because of the largely human-readable format, ASDF can be reasonably inferred even without the specification document. This plays an important role in the data format's applicability as an interchange and archive format.
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HDF5's data model is not flexible enough to represent all the needed structures, particularly generalized World Coordinate Systems (WCS).