A) Advaita Vedanta can be called a mystical path, a spirituality, science of reality, or a combination of both (which I prefer). It can be called nonduality or ‘Monism’ (preferably the first): monism because it takes reality as being One (“without a second”). Nonduality because – though reality is one in essence or ultimately – it presents itself as apparently two: purusha-prakriti, Self- not self, sat-asat, subject-object, Atman-brahman. That apparent dichotomy, as stated, is reducible to the one reality which can be called variously ‘pure consciousness’, ‘the absolute’, ‘sat-chit-ananda’ (being-consciousness-bliss)… the unnamable. Words – language – are secondary, needed to express what is in itself inexpressible. What is inexpressible can be/is a (self) realization of ‘what is’ (anubhava) arrived at by intuition and (Vedantic) reasoning.
From he above it can be seen that Berkeleyan idealism is quite different. One similarity is that both systems deny the existence of an external world, but the idealism of Berkeley retains the validity or reality of minds and ideas. There is no problem with having God as the final ‘arbiter’ or Witness, since this notion or reality is (ontologically) equivalent to ‘the Self’ or pure Consciousness. Towards the end of his life Berkeley came to a position akin to pantheism*, not quite different from Advaita which, as stated, is a mystical, experiential Way (knowing-being), but properly it is not pantheism. In Advaita the apparent multiplicity of forms/objects is denied – they are just names and forms (nama-rupa) , but in the end (with full comprehension) they are not other than the way Consciousness or the Self manifests Itself, thus ultimately not different from It.
* His last work, ‘Siris’ c.f. Colin M. Turbayne’s “Berkeley’s Two Concepts of Mind”
c.f. Greg Goode on Non-duality (Western types of).
B) In Advaita Vedanta Vedantic (or higher) reasoning is distinguished from independent reasoning or speculation, which invariably is in conflict with that of other individuals and schools of thought – ‘Speculation is unbridled… It is impossible to expect finality from it, for men’s minds are diversely inclined’ (SBh 2-1-11). The former, higher reasoning, is, or must be, in agreement with scripture (Upanishads, etc. called shruti) and is never in conflict with universal experience. There is some syllogistic deduction (‘there is fire on that mountain for we see smoke there’), but it is not prominent in AV.
‘For the truth relating to this Reality which is conducive to final release is too deep even for a conjuncture without revelation’ (SBh 2-1-11). Here ‘revelation’ means the ‘deep intuitions arrived at by the sages of old (rishis)’ and compiled in three main bodies of works (chiefly the Upanishads), so you can disregard that word and substitute ‘self-realization’ for it.
But even scriptures are not sufficient to get to the truth: a prepared, mature mind is a requisite, which usually takes years if not life-times. After that long preparation, preferably with the help of a qualified teacher, a final intuition (anubhava or brahmavidya) may occur. I won’t talk about the method or methods used or about qualifications of the student, not a small matter.
‘Right knowledge ought to be uniform throughout, since it must conform to an existing fact. That is to be considered real which consistently maintains its identity… right knowledge, as for instance the knowledge that fire is hot’ (SBh 2-1-11). There is of course much more to say about all this [what is ignorance (avidya) as per AV, means of knowledge, levels of understanding – rather than of existence or being –, etc.].