Suppose you hold the view, often attributed to Saints Augustine and Aquinas, that God is "absolutely simple". What follows from this? Well, it depends, of course, on what you mean by "absolutely" and "simple". It also depends on what you mean by "God" and "is". (In philosophy, everything is a matter of entailments from conceptual starting points.) Suppose what you mean by "God" is the Three Persons of the Christian Trinity; we'll look into those other concepts presently.
To say that something is "identical to its essence" is a metaphysical slogan that goes back to Aristotle. What the Stagirite had in mind is this: a substance, that is, a metaphysical unity or primary being, is something that can be considered under two aspects. On the one hand, the ontological aspect is the substance as such. For Aristotle, for example, a particular man or a particular horse would count as a substance, and each of these things is a material entity that is "informed", if you will, by virtue of belonging to a particular biological species. A particular human is a material manifestation of the species homo sapiens; the species determines what kind of thing the man is, and the matter determines which instantiation of that particular kind he is. From an ontological point of view, the man is a compound of matter and form. But from an epistemological point of view the man is just form: only the universal can be an object of knowledge, and the species is that universal, present in the matter/form compound. When Aristotle says that each thing is identical to its essence, what he means is that to know what that particular man is just is to know the form, the species, to which he belongs. But it should be obvious that there is another sense in which the particular man is not identical in a Leibnizian sense with that form, since the form is immaterial and the particular man is a matter/form compound. So there must be a distinction here between two different kinds of primacy: primacy in the order of ontology (matter/form compound) and primacy in the order of understanding (immanent form, essence).
Matter is pure potentiality, a principle of individuation in most metaphysical systems. God is not only immaterial, he is sui generis, hence there is no principle of individuation associated with the substance that is God. That is precisely why there is only one of him. (Within Christian theology, angels, too, are each of them sui generis, since they are also immaterial and matter is the principle of individuation in Christian metaphysics.) It is essential to note that the Divine Persons are not principles of individuation in God, else there would be three gods, not one God. There is a single, Divine Nature that the three Persons all share. In the case of the particular man, Jesus of Nazareth, there are, interestingly, two different natures compresent: the Divine Nature of the Second Person, and the human nature of the son of Mary. By reason of a peculiar metaphysics, the material, particular entity, Jesus of Nazareth, is identical to the Second Person of the Trinity, even though Jesus of Nazareth and the Second Person of the Trinity do not share all of their properties.
This is an important point to keep in mind. Jesus of Nazareth, the son of Mary, is a temporal being. That is, he came into existence at a particular point in time, and he perdured for a certain extent of time, and then he ceased to exist as a material entity. (This point may be open to dispute, depending upon what is intended in the Scripture by the expression "glorified body"; but at the very least it is true to say that Jesus came into existence at a particular point in time, which is not true of the Second Person of the Trinity.) Traditionally, identity is understood thus:
(x)(y)(F)[if (Fx iff Fy) then (x = y)]
Hence, in order for the Christian metaphysics needed for the doctrine of the Trinity to be compatible with the doctrine of Christology, a different conception of identity is required. My purpose here is not to provide an account of that conception of identity; it is sufficient for my purpose in this post to show that it is a non-standard notion to the extent that it fails to be Leibnizian.
Since the notion of identity required to make any sense out of Christian metaphysics is already a non-standard notion, folks who appeal to identity when making extravagant claims about this or that theological system need to be somewhat circumspect when claiming that identity makes nonsense of the system under consideration. For example, there are some who claim that, because God is what he is by necessity, and because he willed to create, if you think that God is absolutely simple you are committed to thinking that God's will to create was a matter of necessity and, hence, you are committed to the unacceptable view that God was not free when he created. This objection is often put forward as an objection to the Augustinian and Thomistic notions of absolute divine simplicity, though it could just as easily stand as an objection to the necessity of God's essence. Which problem you pick on is, apparently, a function of believing that one of the problems is not really a problem, while the other one is a problem.
There are different sorts of necessity out there. One kind is merely logical. For example, it is logically necessary that all bachelors are unmarried males. That is logically necessary because the meaning of the word "bachelor" guarantees the truth of the universal generalization. Another kind is what we might call "historical necessity". If we define "necessity" as "that which cannot be otherwise", then the facts of history, which cannot now be changed, are, in a very tenuous sense, necessary truths. By contrast, if we define "necessary" as "true in every possible world" (a definition that could satisfy only a 20th century analytic philosopher) we find that there is no such thing as "historical necessity", but there is such a thing as "rigid designators", and we may stipulate that, as a matter of reference, water, for example, just is H2O in every possible world and, hence, water is necessarily H2O (this is a controversial view).
And then there is a sort of necessity that seems to fall somewhere between these other examples. Take the act that is named by the phrase: "sacrifice by means of cutting the throat". This act has a particular consequence, namely, the death of the thing sacrificed. As a matter of physical fact, every thing that is sacrificed in this way dies. As a matter of necessity (if it were to survive, either by miracle or by surgical intervention, it would not be true to say of it that it had been "sacrificed"--even Our Lord had to die in order for his act to count as a sacrifice). Now, it should be obvious that the act named by the expression "sacrifice" is not the same thing, that is, is not identical to, the act named by the expression "die". And yet the act of sacrificing entails, by a kind of necessity, the act of dying. There is a necessary connection between the two without the two things being identical. There might be a question of boundary conditions here: perhaps the slicing of the throat in sacrifice is the "beginning of the process of dying" or some such. But it is possible to get a little too anal in these kinds of cases.
In what sense, then, is God identical to his essence, and what follows of necessity if his essence is a matter of necessity? One thing that does not follow is that every act of God's will is necessitated in the sense that he could not have willed otherwise. This is clear enough simply by considering that God's properties are all of them atemporal, and so "willing to create", if it is to be understood as applying to an atemporal being such as God, must be but one disjunct of a disjunctive property, "willing to create or not to create", only one of which can be realized as a matter of logical necessity. God has this property in precisely the same sense that I have the property "able to write a blog or not to write a blog", and God was as free not to create as I am free to refrain from writing this blog (as I am sure many folks are hoping). Quite obviously God is not identical to this property, any more than he is identical to any of his other properties. To say that God is simple is to say no more than that he is sui generis, it is not to say that he may not be analyzed, as Aristotle was fond of saying, logikôs. It might be fair to say that God is identical to his essence, and that his essence is, as it were, a metaphysical unity of all of his properties, but this conveys no necessity to any particular willing on God's part. At worst in involves us in the puzzle of what counts as identity in Christian metaphysics, but that will be a problem for any Christian, not just the Christian of an Augustinian or Thomistic stripe.