Advaita is usually stated as a proposition, that the self and Brahman are not two. Inside the tradition it is a method before it is a proposition, and the method is what distinguishes it from the schools that make the same noises.
Superimposition and its removal
The technique is adhyaropa apavada: deliberately superimpose a description, then withdraw it. The teacher says Brahman is the cause of the world, and then takes causality back, because a cause requires an effect that is genuinely other. Every positive statement is provisional scaffolding to be dismantled. This is why the tradition can use theistic language freely without being theistic in the way the Vedanta schools that oppose it are.
The great sayings, and what is done with them
Four sentences from the Upanishads are treated as the pivot: that thou art, this self is Brahman, consciousness is Brahman, I am Brahman. They are not mantras. They are propositions to be heard from a qualified teacher, then reasoned through against every objection, and then dwelt on until the doubt no longer arises. Shravana, manana and nididhyasana name the three stages, and skipping the middle one is the standard charge levelled at popular presentations.
What the modern versions changed
The Vedanta that reached the West through Vivekananda and after him is usually called neo-Vedanta, and scholars separate it from the classical school on two counts: it treats direct experience as the criterion where Shankara treats scripture as the criterion, and it presents Advaita as the inner truth of all religions, which is a nineteenth-century apologetic move rather than a classical one.