There's a bunch of odd similarities between them, for one they were both in open relationships, the both wrote novels to convey philosophy, they both used amphetamines, were heavy smokers and died of lung disease.
According to Nathaniel Branden, early in her career (perhaps owing to her earlier Nietzchean influences) she one considered naming her philosophy existentialism, but decided against it.
They both have a very minimalist ontology, limited to a few broad descriptive categories. They both uphold the primacy of existence and a kind of conscious intentionality, that is, that consciousness is awareness of objects, and not simply awareness of itself, and both reject the prior certainty of consciousness and the cogito.
Sartre, however, is a phenomenalist in the tradition of Hegel and Heusserl and so upholds a kind of Kantian thing vs thing in itself distinction, though he does believe in the validity of sense perception, although sorta kinda, because the fact that sense perception is limited and fallible counts against them for him and not for Rand. Plus Sartre days a bunch of incomprehensible gibberish like, "consciousness is nothingness," which Rand denies the possibility of.
Apart from that they both stridently believe in free will, but Sartre's is a kind of indeterminist and acausal agency that overrides and literally cancels out the causal reality that underlies it("nihilation".) Rand's is compatible with the law of causality and is a naturalistic faculty at one with biological identity. They both also draw different conclusions, for Rand volition is the startig point of human value achievement, and so undergirds her heroic and optimistic ethical egoism, whereas Sartre pessimistically laments free will, which "condemns" us to make choices and face suffering and failure and navigate a nauseous array of subjective values.
Sartre, in general agrees that reason and science are valid and efficacious, but are cold impersonal, only giving us formal knowledge, but not meaning and purpose in life. But they're both atheists and are searching for meaning and purpose and want to substitute a kind of secular humanism in the place of religion.
Sartre has a lot (more) to say about psychology as well, but I'll cut it short there.
1. They believe in an Aristotelian conception of a soul as a natural faculty of biology. A soul or consciousness is a capacity of certain animals' neurological systems which gives it motor functions and awareness, as well as a selective focus and, in humans, the ability to abstract and form concepts and language.
2. A common argument of determinists is that since free will is conceived of in a Platonic or religious manner, in order for free will to be valid, it would have to be a magical or infinite. This is called libertarian free will. The law of identity certainly does refute this type, but not a naturalist version. Free will, like vision or hearing, being a biological function, is dependent on organs, is finite and limited.
3. Again, much like the same dichotomy, we can either have moral relativism on the one hand or a substantive, but mystical morality. But in an Aristotelian-Randian conception, morality isn't random emotiveness or appeals to the supernatural, it is common sense principles for achieving a good life and well being. Since man is a being with a specific nature, and that nature is governed by laws knowable by rational inquiry and investigation, just like say a tree or an elephant, humans are capable of investigating the conditions and principles necessary for survival, continued growth and success of living entities. It is, again, a naturalistic view of ethics.
Objectivism doesn't name anything so concrete.
If you want a broad list of Objectivist principles, there are four points listed on this page (scroll down a bit): https://campus.aynrand.org/works/1962/01/01/introducing-objectivism
If you limit the question just to "what we should value", then here's a quote from Rand:
See a little more here: http://wiki.objectivismonline.net/Virtues