It seems that from this question that the Advaita Vedantic school of Adi Shankaracharya has two divisions- the Vivarana school of Prakasatman and the Bhamati school of Vachaspati Misra. What are the philosophical differences between these schools other than the nature of Avidya as described in that question?
There are as many as six differences I could find and comprehend, between the Bhamati school of Vachaspati Mishra and Vivarana school of Prakasatman:
• The Bhamati school opines that the purpose of Karma is to inspire one to desire for knowledge. By performing one's duties efficiently without expectating its fruits and offering it to Ishwara, the desire for knowledge is arises in an individual.
• The Vivarana school opines that by performing one's duties efficiently without expecting it's fruits, knowledge automatically arises in an individual.
• Bhamati school believes that, in order for an individual to develop detachment towards materialism, one must learn to discriminate between what is reality and what is unreality. It is not enough to know that world is not eternal since it does not produce detachment because they still generate happiness.
• Vivarana school believes that an individual can detach by simply knowing the non-eternality of the world.
Study of scriptures
• The Bhamati School stresses on the understanding of scriptures for all everyone, regardless of whether they want to practice Karma-Yoga or Jnana-Yoga.
• The Vivarana School opines that rote learning of scriptures is sufficient for people who wish to practice Jnana-Yoga as the understanding of the meaning of the scriptures is automatically attained through Guru's grace.
Avidya in all individuals
• Bhamati school believes that the Avidya in an individual differs from person to person.
• Vivarana school believes that Avidya is the same in all individuals.
Nididhyāsana and Shravana
• Bhamati school stresses on the importance of Nididhyāsana or profound meditation on the Upanishadic Mahavakyas.
• Vivarana school stresses on the importance of Shravana, the hearing of Vedantic secrets from one's Guru.
Cause of Brahmajnana
• Bhamati school believes that Manas is the main cause of Self-realization in an individual.
• Vivarana school believes that the Upanishadic Mahavakyas are the main cause of Self-realization in an individual.