TITLE: $ \{x : P(x)\} $ vs. $ \{P(x) : x\} $ ---- When are these set-builder notations the same and different?
QUESTION [7 upvotes]: I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer.  
From Gary Chartrand page 216 Mathematical Proofs - 
$\begin{align} \text{ range of } f & = \{f(x) : x \in domf\} = \{b : (a, b) \in f \} \\ 
& = 
\{b ∈ B : b \text{ is an image under $f$ of some element of } A\} \end{align}$
Wikipedia - $\begin{align}\quad \{\text{odd numbers}\} & = \{n \in \mathbb{N} \; :  \; \exists k \in \mathbb{N} \; : \; n = 2k+1   \}     \\
& = \{2n + 1 :n \in \mathbb{Z}\} \end{align}$
But  Why $G/G = \{gG : g \in G \} \quad ? \quad$ And not $\{g \in G : gG\} ?$
EDIT @Hurkyl 10/5.  Lots of detail please.
Question 1. Hurkyl wrote $\{\text{odd numbers}\}$ in two ways.
But can you always rewrite $\color{green}{\{ \, x \in S: P(x) \,\}}$ with $x \in S$ on the right of the colon? How?
$ \{ \, x \in S: P(x) \,\} =  \{ \, \color{red}{\text{ What has to go here}}  : x \in S \, \} $? Is $ \color{red}{\text{ What has to go here}} $ unique?
Qusetion 2. Axiom of replacement --- Why $\{ f(x) \mid x \in S \}$ ? NOT $\color{green}{\{ \; x \in S \mid f(x) \;
 \}}$ ?
@HTFB. Can you please simplify your answer? I don't know what are ZF, extensionality, Fraenkel's, many-one, class function, Cantor's arithmetic of infinities, and the like.

REPLY [2 votes]: Mauro makes part of this point, but it's worth stressing: the two different ways of notating a set are used for instances of application of two different axioms (strictly, axiom schemas) in ZF: 
$$
B = \{x \in A : \phi(x)\}
$$
is an application of the subset axiom (a.k.a. separation):
$$\forall y_1, \ldots, y_n \forall A\, \exists B\, \forall x (x \in B \leftrightarrow x \in A \wedge \phi(x, y_1, \ldots, y_n, A)),
$$ where $\phi(x, y_1, \ldots, y_n, z)$ is a formula in the language of set theory. The symbol within the curly braces is guaranteed by this axiom to refer to an element of the domain of a model of ZF (and to exactly one element by the axiom of extensionality). So thanks to this axiom it is always a legitimate shorthand notation for a set.
Conversely, $
B = \{f(x): x \in A\}
$
is an application of the Fraenkel's axiom (schema) of replacement. The axiom schema is
$$\forall y_1, \ldots, y_n, \forall A \left(\left(\forall x (x \in A \rightarrow(\forall z_1, z_2\, (\phi(x, y_1, \ldots, y_n, A, z_1) \wedge \phi(x, y_1, \ldots, y_n, A, z_2))\rightarrow z_1 = z_2\right)) \rightarrow \exists B \forall z (z \in B \leftrightarrow \exists x (x \in A \wedge \phi(x, y_1, \ldots, y_n, A,z)))\right)
$$
What this says is that if $\phi(x,z)$ is many-one when restricted to the domain $A$, so that there is a function (in the most abstract sense, i.e, a "class function") satisfying $x = f(z) \leftrightarrow \phi(x, z)$, then the collection of all the images of elements of $A$ under $f$ is also a set. 
Or, in shorthand, there is a set which is the elementwise-image of $A$ under $f$. Again, by extensionality there is exactly one such set.
So which of the two notations you use for a particular set would be defined by the last axiom you apply in proving the existence of that set: are you thinking of it as a subset of something, or as the image of something? It should be obvious that $\{x \in \mathbb{N} :x\text{ is even}\}$ is the same set as $\{2\cdot x: x \in \mathbb{N}\}$. So very often the notation you choose will be determined by what you are wanting to say about a set that could be constructed either way. And as others have noted you can roll the two notations into one, when a set is constructed by successive applications of both axioms, and save paper.
You cannot always eliminate Replacement in favour of Separation: ZF is strictly stronger than Z (the collection of axioms omitting replacement). In general Replacement is needed to conduct Cantor's arithmetic of infinities inside set theory: see the Wikipedia page on replacement. A specific set that cannot be proved to exist without replacement is $\aleph_\omega$. But if you know a set $A$ exists you can trivially write it in either form with a bit of shorthand: $A = \{x \in A : x = x\}$ and also
$A = ${$\text{Id}(x): x \in A\}$ (where $\text{Id}$ is the identity map). I hope this answers Question 1 of the edit.
For Question 2, note that your proposed backwards replacement set notation is indistinguishable from the usual subset notation, unless you are absolutely rigorous in separating your capital $P$ shorthand for a formula from your lower-case $f$ for a function. There would be an ambiguity in the notation when $P(x,y)$ had an extra free variable: is it the formula defining a function so the set is constructed by replacement, or is it being used to define a subset (for any given value of the free variable)? It's a bad idea!
More intuitively, you can pronounce $\{P : Q\}$ in English as "the set of elements P such that Q", and if you start reversing some of the notation it becomes much harder to read. 

It's also worth noting that taking subsets, and taking the image under a function, are entirely intuitive operations that you want to do naively with sets well before axiomatising them: given a classroom of children, the set of girls in the classroom and the set of all fathers of the class are collections that any teacher might need to think about. The notation makes sense without worrying about the axioms.