The heyday of existential Thomism is well past, but Etienne Gilson and others were certainly not wrong to take note of a close connection, despite other large differences.
Heidegger in Being and Time (1926) famously claimed that philosophers since Plato had been preoccupied with questions about beings and had lost sight of the central importance of Being writ large. Many 20th century Thomists partially accepted this argument, but contended that Aquinas was an obvious exception, citing Aquinas’ identification of God with pure Being. Heidegger rejected that identification, and would have insisted that Being was not a being at all, not even the unique one in which essence and existence were identified. Nonetheless there is a broad parallel, to the extent that Heidegger and Aquinas each in their own way stress the dependency of beings on Being.
In some circles, Aquinas has been criticized for promoting a “philosophers’ God”. But according to Burrell, Aquinas argued in effect that on the assumption that there is only one God, the God of Summa Theologica and the God of common doctrine must be acknowledged to have the same referent even if they have different senses, like Frege’s example of the morning star and the evening star.
Sartre in his 1945 lecture “Existentialism is a Humanism” put forth the formula that “existence precedes essence”. Aquinas in Being and Essence had argued that God has no essence other than existence. Sartre argued in effect that the human has no essence other than existence. In his context, this is to say either that the human essence consists only in matters of fact, or that there is simply no such thing as a human essence.
Sartre’s use of the word “essence” reflects a straw-man caricature of bad essentialism. Whatever we may say that essence really is, contrary to Sartre’s usage it is supposed to be distinguished from simple matters of fact. On the other hand, in formal logic, existence does reduce to matters of fact.
What Aquinas, Heidegger, and Sartre have in common is that they all want to treat existence as something that transcends the merely factual and formal-logical. Speaking schematically, it is rather the analogues of essence that transcend the merely factual in the Platonic and Aristotelian traditions. Thus Aquinas made a major innovation in inventing a new, unprecedented concept of existence that transcends the factual. I’m inclined, however, to sympathize with Dietrich of Freiberg’s argument that the concept of essence could already do all the work that Aquinas’ new supercharged concept of existence was supposed to do.
What is important for practical purposes is that there is something that transcends the merely factual. I think the close connection of “essence” with form and ends makes it an ideal candidate. The big difference between form and ends on the one hand and facts on the other is that logically speaking, facts can be arbitrary, whereas any form or end or essence necessarily implies some nonarbitrary order.
For Aquinas, God is simultaneously a fact and more than fact, and is unique in this regard. Nothing else has this dual status. Sartre transferred this unique dual status to the human. By contrast, the neoplatonic One is strictly more than fact — in traditional language, the One as source of being was said to be “beyond being” altogether. The 20th century theologian Paul Tillich quipped that it could be considered blasphemy to say that God exists (because “existence” is mundane and factual).
The “To-Be itself” of Aquinas, while profoundly innovative with respect to previous tradition and certainly not strictly Aristotelian, is nonetheless arguably more Aristotelian in spirit than the neoplatonic One, insofar as it is less ambiguous about the goodness of the actual world. Plotinus struggled mightily to reconcile a commitment to the goodness and beauty of this actual world with an ascetic tendency to devalue all finite things in face of the infinite One. In Aquinas there is still some tension between the reality of secondary causes and the absolute dependence of everything on God, but I think it is fair to say that the way Aquinas sets up the problem makes the reconciliation easier to achieve. This was a huge accomplishment. Nonetheless, taking into account other factors like assertions about the place of omnipotence and sheer power in the scheme of things, my overall sympathies lie more with the neoplatonic “strictly more than fact” perspective, and even more so with Aristotle’s more modest view that the “First” cause is strictly a final cause.