With regard to ontology, I accept many of her views especially related to Aristotelian logic (but not her rejection of Divinity), e.g. one objective reality independent of human sense-perceptions, the law of non-contradiction, the law of the excluded middle, the law of identity, the primacy of the concept, the motive power of the Idea, &c. I have a profound distrust of people who question such basic principles as objective reality or the law of non-contradiction, because it shows that they are either unintelligent or reflexively contrarian. I don’t believe anyone can really believe A∧¬A.
With regard to epistemology, I differ with her in generally rejecting reason as the primary means of knowledge (obviously not wholly rejecting reason as a means of knowledge), placing intuition far above reason in the epistemological hierarchy. I do not believe that human reason in practice functions as a reliable chain of logical deductions from sound præmises, as we can observe from the fact that rationalists tend to make the least rational decisions in conducting their lives. I see reason as a means of making sense of and consolidating higher means of knowledge, namely revelation (the highest) and intuition (the second-highest).
With regard to morality, I see many elements of rational egoïsm in Vedic mōrēs. Selfishness is a good thing, and actions like sacrificing your life for a loved one are not altruistic if you love them so much that they rank higher than you in your hierarchy of values. Sacrificing your life for a random stranger unrelated to you is an act of altruism and is deeply evil. In the Vedic religion there is (with rare exception) no altruism, but a selfishness of clan in which one’s own kin matter the most. The r̥ṇá-s that you must fulfill as the goals of life are also egoïstic in nature: propagating your own lineage, achieving Heaven for yourself and your wife, strengthening your Fathers in Heaven, achieving wealth and health and glory for yourself and your family. The yajñá exists to strengthen the Gods because They created you (gratitude) but also for reciprocity (dehí me dádāmi Te) so that you can benefit from Their favor.
All this doesn’t præclude charity and kindness to people who aren’t your kin—nor does Rand. Well, one could argue that Rand has a nearly universally negative view of charity, in the sense that even charity between friends is portrayed as dishonorable. Here her view would differ from the Vedic.
With regard to politics, I disagree with Rand on virtually everything. I believe that freedom is an important and Aryan virtue, and in principle I align with libertarians on a few issues (e.g. “taxation should be lowered”), some very strongly like gun rights. But her political philosophy is founded on individual rights, a concept that literally does not exist in the Vedic or any other religion. Obviously family and society are more important than individual rights. So obviously you don’t have the right to cheat on your husband or get an abortion or not pay taxes. It’s quite ludicrous when people describe Rand as “right-wing”; in any way that really matters, she’s extremely leftist.
But I agree with Rand in her largely herrenmoralisch view of mankind (even if she emphatically rejects Nietzsche, there are strong resemblances). The good guys really are a tiny fraction of humanity, the people with real minds who think and feel like men were intended, who are meant to, and have every right to, rule over the honorless and slavish masses. Vedically a small élite of Brahmá–Kṣatrá rules over the common Víş and expropriate wealth from them at will; and the Brahmá–Kṣatrá–Víş, that is, the Ā́rya-s, are together an élite meant to command other men, the Şūdrá-s. Of course Rand would disagree with the idea that the true spiritual élite rule by force, since she is a Víş supremacist.
To give a fully detailed answer to your question would probably require writing a long essay or book, which I actually do plan to do but only years from now.