“The Great” is a title that the 2 groups see in meaning, completely different. That’s why the disagreement.
The Conciliar group, if I may call it such, sees Great:
inclusion of all religions in salvation
ecumenism in the modern revised sense
extreme inculturation of the liturgy.
Insulation from responsibility for the unprecedented advancement of novel liberalism.
The traditional group sees
Explicit promotion of the Catholic religion to non-catholics; a consistent mark of historical popes
advancement or promotion of traditional hierarchy vs promotion and maintenance of liberal if not morally corrupted hierarchy
strict conformity in the liturgy
no compromise to novelty in the Mass because of liberal influences
conciliar rule in the church giving loss of leadership to the pope in faith, morals and praxis.
no discipline offered to most of the heterodox teachers corrupting the traditional catholic faith.
Apparently ignoring the need to root out on an active basis, the sexual predators in the seminaries.
In other words playing the fiddle for the amusement of the crowds while the church is burning down.
In summary, the word Great has 2 different meanings for each group.
One, the greatness of world popularity.
The other as the maintenance of the Catholic religion in all it’s aspects.