I have discussed this in other answers I have given here, so see some of my other posts. But it is an excellent question, and some answers bear repeating for the sake of clarification.
As I have said elsewhere to be briefly touched upon here. Being only exists in Indo-European languages, and existence came into our lexicon as we translated Arabic texts on Aristotle back into Latin. We had to create a technical term for wajud in Arabic and so we produced existence, which means exi-stance, to stand outside of, and also ecstasy. Wajud really means what is found. Existence is what stands outside of Being, prior to the manifestation of Being, but also manifests under the veil of illusions that are created by Being. Heidegger, used the idea of ecstasy to say that dasein is the one who projects Being as a being-in-the-world as an ecstasy, thus dasein throws Being into being as becoming yet at the same time is thrown into the world. Generally it is thought that Sartre got Heidegger wrong and his thought has been devalued recently by critics, as a bad imitation, and thus Critique of Dialectical Reason is considered his greater work in terms of originality, even though his most popular work was and probably still is Being and Nothingness. Merleau-Ponty saw nothingness as being the inverse dual of Process Being (ready-to-hand modality of Being). Thus he calls the third meta-level of Being the Hyper-dialectic between Being and Nothingness which I shorten to Hyper Being and which Derrida called Differance (differing and deferring) and which Heidegger called -B-e-i-n-g- (crossed out). However, we can see that Sartre was reconsidering Heidegger in a Hegelian context. And if we note that Heidegger begins and ends Being and Time with references to Hegel, and that Heidegger uses the term dasein, which for Hegel in his Logic was determinate being, and the German philosophical term for “existence”, then perhaps this critical view perhaps needs to be reassessed.
As I have elaborated elsewhere, the concern for existence was an undercurrent of philosophy since the Renaissance. Essence concerns Being as it manifests as kinds of things. Existence concerns what is found. Eventually existentialists realized it could be used as a way to reverse things within Western Metaphysics to say that Existence precedes Essence, rather than the other way around which is basically an idealist approach to things which is endemic to our worldview. Existence points toward the concrete manifestation of singular individuals that are found, rather than to their constituent attributes and the constraint ranges on them which determine their species or kind. For instance Maturana and Varella’s Autopoietic theory is existential biology because they concentrate on the viability of the individual organism, rather than their connection to a species as an exemplar. The move toward existence as an important concept appears in Dostoevsky Kierkegaard, Heidegger, Merleau-Ponty and others following in their footsteps. Sartre played an important role in the development of Existentialism, and by his formulation of it as existence preceding essence, rather than what is normally believed which is that they are another way around. This is also seen as supporting the Marxist position that material things are important, without being quite as crass in its explication.
But in both Heidegger and Sartre, they do not achieve Existence as is mean originally by the term wajud. Neither actually escape Being, they merely manage to go up a meta-level to the next higher stage of Being. Sarte shows this in his terminology taken from Hegel of being-for-itself, being-in-itself, and being-in-and-for-itself. Heidegger shows that because the ecstasy of dasien can be seen as a form of becoming. For them Being is essentially Being taken to its limits, not beyond its limits to a new standing.
The development of Existence as a mode for critiquing Being, in terms of Process Being and which considers Pure Being to be identical with essences was a big advance in metaphysics, and it goes back to the distinction in Husserl between Abstraction (Pure Being) and essence perception (Process Being). Note essence perceptions which is a transformative operation produces representations of essences which are constraints on attributes of a thing. Those representations are presented in Pure Being. So there is a circle here of representation which is repeated. Heidegger talks about this in relation to the fact that Dasein is within the world he projected ecstatically. A fundamental contradiction of the order of the idea of Christ who is God in the world that God as Father created to whom the Holy Spirit came who is by the way also God. Heidegger does not talk about essences because he says that these belong to entities other than Dasein to which the Categories of Kant apply. But instead we need to recognize the existentalia which are the same as the categories for humans of which there are three (verstehen, rede, befindlichkeit). That is understanding, talk and discoveredness whose overlapping give us Care.
Sartre is coming to the whole issue from a Marxist perspective, and thus for him existentialism is the new way to understand the materialism of Marx from a psychological perspective. Marx himself did not question the preeminence of essence over existence, because existence was not on the radar. Material things were thought to have essences. Existentialism made materialism into a psychological state and explained some of the problems of Marxist ideology which was normally thought about in very mechanistic ways. Existentialism was the way that things looked from the point of view of the individual. And it meant a difference in the way the individual fit into the world related to its absolute freedom, because the individual made the world, he made the meaning of his world, and therefore he made himself. Existentialism of both Sartre and Heidegger was very individually oriented. An excellent book that makes this point is Existence and Love by Wm. Sadler. Sadler makes the point that phenomenology is very visually oriented and this makes it concentrate on the individuals experience, whereas if the major mode of the senses was auditory then the problem of intersubjectivity that plagued Husserl and that Heidegger set out to solve would not have existed, because sounds interpenetrate and are not seen as something distant and separate. The fact that existentialism concentrates on the individual as something perceived, something under the gaze as the other, means that it always turns out to be rather bleak, as we see in Camus, or for instance in Sarte’s waiting room where Hell is other people. By using Sadler we can get a handle on what Heidegger and Sarte have in common as a fundamental assumption. Sartre is basically reversing Heidegger and looking at Nothingness as the active groundlessness of consciousness itself. This is the inverse dual of Heidegger’s Process Being (ready-to-hand, grasping) which is a modality of the Monolith of Being just as is Pure Being (present-at-hand, pointing). Being is also groundless for Heidegger and that shows up in its endless becoming, but that becoming leads to the projection of representations that are present-at-hand, i.e. that we can point at. But for Sartre consciousness is groundless in a way that it is always involuting and thus falling into nothingness, so nothing can stand in consciousness for very long, and thus we must produce what ever meaning we can appeal to ourselves. For Heidegger Being is itself intelligibility, and as Parmenides says “Being and thinking are the same”, and thus the process of projection is itself positive in that it produces though transformations and processes the static representations as products. Sartre on the other hand recognizes something that Heidegger misses. Heidegger posits dasein as what is prior to the subject/object dichotomy. But Heidegger more or less forgets about the object, which we might call the eject, i.e. the proto-object, perhaps something like the placenta that comes into being with dasein yet is not either an object nor a subject. Sartre adopts the vocabulary of Hegel and talks about in-itself, and for-itself. Thus there are the things that are not seen as part of the ego which are in-themselves, and these are different from what the ego recognizes in the moment which is for-itself which is pure freedom and pure reflectivity. There is also being-for-others which is always a temptation. And finally there is the pure freedom of God for which man strives but which is the reflectivity of reflectivity which is being-in-and-for-itself. Thus man can get lost in material things (in-itself), in others (for-others, which Heidegger calls mitsein) and in the in-and-for-itself of the reflectivity of reflectivity of God which is an unsustainable limit. Man is only truly himself if he has the practical freedom of the in-itself which is reflective and which allows him to see himself in the world as its maker, and also as the made, by his own actions.
In Heidegger, the emphasis is not on reflectivity, but on the ecstasy of dasein pouring forth Being from the inside, which then allows him, to be-in-the-world that is spun from out of him. In Sartre the emphasis is on reflectivity, which is Hegelian self-consciousness (for-itself) and there is a limit in the Absolute of squared reflectivity which reminds us of pure Spirit that is in-and-for-itself. Sartre’s for-others (mitsein) is collapsed into Spirit in Hegel. Spirit is the Social polis pursuing is historical destiny. Since Hegel is an idealist for him the in-itself does not really exist completely, but that is exactly what makes the Marxist reversal of Hegel possible. But Marxism in its pure form causes Nausea because it means the things of the world are too much with us. When one denies one’s freedom and acts as if one is an object that is bad faith for Sartre, a lack of authenticity in which one forgets the for-itself and its freedom given through reflectivity. One can also be inauthentic by living for others, and trying to be what they want you to be instead of being yourself and exercising one’s inherent freedom. One can also think of oneself as god and believe that one can remake everything in ones own image and thus be caught in the in-and-for-itself the limit of human existence at the higher meta-level. So for Sartre the only way to be authentic is to avoid these three pitfalls and realize ones being-for-itself and realizable freedom to be authentic.
A good book that gives another take on this from another perspective is Being-In, Being-For, Being-With by Clark Moustakas.