Sri Aurobindo
Letters of Sri Aurobindo
7. The Purpose of Avatarhood
An Avatar, roughly speaking, is one who is conscious of the presence and power of the Divine born in him or descended into him and governing from within his will and life and action; he feels identified inwardly with this divine power and presence.
A Vibhuti is supposed to embody some power of the Divine and is enabled by it to act with great force in the world, but that is all that is necessary to make him a Vibhuti: the power may 
be very great, but the consciousness is not that of an inborn or indwelling Divinity. This is the distinction we can gather from the Gita which is the main authority on this subject. If we follow this distinction, we can confidently say from what is related of them that Rama and Krishna can be accepted as Avatars; Buddha figures as such although with a more impersonal consciousness of the Power within him. Ramakrishna voiced the same consciousness when he spoke of Him who was Rama and who was Krishna being within him. But Chaitanya’s case is peculiar; for according to the accounts he ordinarily felt and declared himself a bhakta of Krishna and nothing more, but in great moments he manifested Krishna, grew luminous in mind and body and was Krishna himself and spoke and acted as the Lord. His contemporaries saw in him an Avatar of Krishna, a manifestation of the Divine Love.
Shankara and Vivekananda were certainly Vibhutis; they cannot be reckoned as more, though as Vibhutis they were very great.