Although my personal experience has been limited to observing some rather intense debates between traditional advaitins and neo-advaitins, this differentiation seems to have been extended to vedantists in general and by further extension, to hinduism itself. So, it would be good to understand the differences at the different levels.
Western Indologists consider Rammohun Roy, Keshab Chandra Sen, Ramakrishna, Vivekananda and Radhakrishnan as exponents of different varieties of Neo Vedanta. I will only discuss the views of Ramakrishna, Vivekananda on some issues and to a smaller extent Radhakrishnan.
Ramakrishna and Vivekananda
Vedas are not proof. Anubhav or aparokshanubhuti obtained through sadhana is the proof.
"There are certain religious facts which, as in external science, have to be perceived and upon them religion will be built. Of course, the extreme claim that you must believe every dogma of a religion is degrading to the human mind. The man who asks you to believe everything, degrades himself, and, if you believe, degrades you too. The sages of the world have only the right to tell us that they have analyzed their minds and have found these facts, and if we do the same we shall also believe, and not before. That is all there is in religion."
The Complete Works of Swami Vivekananda, Volume 2, Jnana Yoga, Realisation.
"The proof, therefore, of the Vedas is just the same as the proof of this table before me, pratyaksa, direct perception. This I see with the senses, and the truths of spirituality we also see in a superconscious state of the human soul."
The Complete Works Of Swami Vivekananda, Volume 3, The Sages of India
Attitude towards Smritis and Puranas
Smritis and Puranas are followed by traditional Vedantists.
Such a God I have seen in my life, and his commands I live to follow. The Smritis and the Puranas are productions of men of limited intelligence and are full of fallacies, errors, the feelings of class and malice. Only parts of them breathing broadness of spirit and love are acceptable, the rest are to be rejected.
The Complete Works of Swami Vivekananda, Volume 6, Epistles – Second Series, CXXIV
Attitude towards Sankara’s Advaita Vedanta and Ramanuja’s Vishistadvaita Vedanta
Traditional Vedantists follow either Sankara's Advaita or Ramanuja's or Madhva's or the Vedanta ofother famous Acaryas. Given below is what Ramakrishna says on this issue.
MASTER: "Again, I cannot utter a word unless I come down at least two steps from the plane of samadhi. Sankara's Non-dualistic explanation of Vedanta is true, and so is the Qualified Non-dualistic interpretation of Ramanuja."
The Gospel of Sri Ramakrishna, With the Devotees in Calcutta, March 11, 1885
Is the universe unreal?
Traditional Advaita Vedanta says that the world is unreal. I post below Sri Ramakrishna's teaching.
Master (to M): ”Why should the universe be unreal? That is a speculation of the philosophers. After realizing God, one sees that it is God Himself who has become the universe and all living beings.
“The Divine Mother revealed to me in the Kali temple that it was She who had become everything. She showed me that everything was full of Consciousness. The Image was Consciouness, the altar was Consciousness, the water-vessels were Consciousness, the door-sill was Consciousness, the marble floor was Consciousness - all was Consciousness.
I found everything inside soaked, as it were, in Bliss - the Bliss of Satchidananda. ..........
“After realizing God, one sees all this aright - that it is He who has become the universe, living beings, and the twenty four cosmic principles. But what remains when God completely effaces the ego can not be described in words. As Ramprasad said in one of his songs, 'Then alone will you know whether you are good or I am good!'
A man sees one way through reasoning and in an altogether different way when God Himself shows it to Him.
The Gospel of Sri Ramakrishna, With the Devotees at Dakshineswar (II), December 16, 1883
He was very critical of Gaudapada’s theory of Ajatavada, a version of Advaita Vedanta. I give below an example of him attacking a version of Advaita Vedanta.
The general idea, pervading Gaudapada’s work, that bondage and liberation, the individual soul and the world, are all unreal, makes the caustic critic observe that the theory which has nothing better to say than that an unreal soul is trying to escape from an unreal bondage in an unreal world to accomplish an unreal supreme good, may itself be an unreality. It is one thing to say that the secret of existence, how the unchangeable reality expresses itself in the changing universe without forfeiting its nature, is a mystery, and another to dismiss the whole changing universe as a mere mirage. If we have to play the game of life, we cannot do so with the conviction that the play is a show and all the prizes in it mere blanks. No philosophy can consistently hold such a view and be at rest with itself. The greatest condemnation of such a theory is that we are obliged to occupy ourselves with objects, the existence and value of which we are continually denying in theory. The fact of the world may be mysterious and inexplicable. It only shows that there is something else which includes and transcends the world; but it does not imply that the world is a dream.
Indian Philosophy, The Advaita Vedanta of Samkara, Gaudapada and Buddhism VIII by S. Radhakrishnan