In classical logic this is the same thing. This is a direct result from Stone's Representation theorem, which says that every Boolean algebra is isomorphic to a field of sets, where $\lnot$ is complement; $\land$ is $\cap$ and $\lor$ is $\cup$.
Since in classical logic we mainly deal with Boolean expressions, this should hint you enough.
When considering FOL (where free variables apply) and a structure $M$ one can consider $\varphi(x)$ to be a formula, then $\{a\in M: M\models\varphi(a)\}$ is empty if and only if $\forall x\lnot\varphi(x)$; and the set is $M$ if and only if $M\models\forall x\varphi(x)$.
Indeed this is a good way to think about the formulae, as subsets of the universe. In such case it is easy to see that $\land$ is $\cap$ and $\lor$ is $\cup$.
The infinite case is slightly more complicated since classical logic does not permit infinitary disjunctions or conjunctions. If one allows that, then the same reasoning as above applies.
Indeed what is the meaning of $x\in\bigcup_{n=0}^\infty A_n$? It means that for at least one $n$ we have $x\in A_n$. If $A_n =\{a\in M : M\models\varphi_n(a)\}$ then $x\in\bigcup A_n$ is to say that $M\models\left(\bigvee_{n=0}^\infty\varphi_n\right)(x)$. It is important to distinct between things we do within the language (i.e. formulae we can write) and things we do in the meta-language (i.e. things we know are true due to "higher" reasonings).
Once you have a syntax for your formal logic, you know what the rules are for putting together symbols in your language, so you know what the valid strings are.
Next we need to know what
what the strings mean and
what we can do with these strings.
The first one is given by the satisfaction relation $\models$, which tells you when a string is satisfied, or in other words, when a string is "true".
The second one is given by the rules of inference of the system, also called the axioms, deduction rules, or proof calculus. These describe the relation $\vdash$, and tell you what are the valid rules for proving new statements in the logic. This is the axiomatization of the logic.
So on the one hand we have a notion of truth and on the other hand we have a notion of provability, and of course we would like these two to correspond to each other, i.e. we want the system to be sound (meaning that if something can be proven, then it is true) and complete (meaning that if something is true, then it can be proven).
So to axiomatize a logic means to start with a formal language for your logic and a satisfaction relation that tells you what those strings mean, and from this create a set of axioms or deduction rules such that the logic is at the least sound, and hopefully also complete.
Best Answer
The terms "postulates" and "axioms" can be used interchangeably: just different words referring to the basic assumptions - the "building blocks" taken as given (assumptions about what we take to be true), which together with primitive definitions, form the foundation upon which theorems are proven and theories are built.
The choice to use one particular term rather than the other is largely a function of the historical development of a given branch of math. E.g., geometry has roots in ancient Greece, where "postulate" was the word used by the Pythagoreans, et.al.
So it's largely a matter of history, and context, and the word favored by the mathematicians that introduced or made explicit their "axioms" or "postulates." "Postulate" was once favored over "Axiom", with the development of analytic philosophy, particularly logical positivism, the term "axiom" became the favored term, and its prevalence has persisted since. Perhaps I should be corrected: As Peter Smith points out, it is "more likely" that the "uptake" of the term "axiom" can be attributed to "mathematicians like Hilbert (who talks of axioms of geometry), Zermelo (who talks of axioms of set theory), etc.".
Neither term is more formal than the other. I personally prefer "postulate" over "axiom", since a "postulate" transparently conveys (or connotes - as in connotation) that what we are calling a postulate is "postulated" as a "supposition", from which we agree to work in building theorems or a theory. In contrast, to me, the connotation of an "axiom" is that of a "law" of some sort, which MUST be followed or MUST be true, though it is no stronger than, or different from, a postulate. But again, this is simply a personal observation and preference, and the term "axiom" seems to have more "uptake" at this point in history.