As a follow-up to this related question, I'd like to know under what circumstances, if any, $\Delta x$, $\delta x$ and $dx$ all mean the same thing, and under what circumstances they can all be said to be approximately equivalent in a reasonably valid way.
For bonus points, it would also be nice to know where the inexact differential, $\text{đ}x$, can be used in place of one of the other symbols.
The reason that I ask this is that all of these symbols are commonly used as notation in thermodynamics and stat mech textbooks, and are often interchanged in ways that can be confusing to follow. In the math used in physics it's common to elide exactly correct statements with approximately correct ones, and it would be nice to have guidance when it comes to picking this sort of thing apart.
These letters are frequently used informally.
One thing to keep in mind, though, is that $\mathrm{d}x$ and $\Delta x$ generally refer to opposite sorts of things, although unfortunately the distinction is not often made clear.
$\mathrm{d}x$ is a gadget that tells you how the variable $x$ varies; in particular, it is not a number. If we have a relationship between three variables $z = f(x,y)$, then we know that we can express how $z$ varies in terms of how $x$ and $y$ vary:
$$\mathrm{d}z = f_1(x,y) \mathrm{d}x + f_2(x,y) \mathrm{d}y$$
$\Delta x$, however, tends to refer to a specific variation in $x$. This is actually a bad notation since it isn't just about $x$; e.g. if we are using $x,y$ coordinates on the plane, $\Delta x$ refers to a 'displacement' in which $x$ varies while $y$ is held constant. But if we were using $x, \bar{y}$ coordinates on the plane (where $\bar{y} = x + y$), then $\Delta x$ would refer to a 'displacement' in which in $x$ varies while $\bar{y}$ constant. These are two very different directions to be displaced.
But let's assume we have that problem sorted out. $\Delta x$ often means an actual difference in the value of $x$ at two different points. Going back to the previous example of $z = f(x,y)$, we have the differential approximation
$$ \Delta z \approx f_1(x,y) \Delta x + f_2(x,y) \Delta y$$
Pay attention to the difference between the previous version: the previous one had nothing to do with an actual displacement: it is describing a feature of how variations in $x,y,z$ are related in all possible ways they are allowed to vary. This one, however, it's an approximation between actual numbers coming from a single displacement.
Sometimes, $\Delta x$ means a differential geometry-style infinitesimal: i.e. a (tangent) vector (whereas $\mathrm{d}x$ would be a covector in this style; i.e. a vector of the opposite variance). In this case, $\Delta x$ and $\mathrm{d}x$ can be combined together to produce the number $1$, and $\Delta x$ combined with $\mathrm{d}y$ produces zero (assuming one particular way of sorting out the ambiguity of what $\Delta x$ implies we're supposed to be holding constant). It is more common, however, to use notations like $\frac{\partial}{\partial x}$ (or $\partial_x$) in this setting, though.
This notation is suggestive, because we can write equations like
$$f(P + \Delta P) = f(P) + f'(P) \Delta P$$
$P + \Delta P$ is a suggestive way to refer to a particular vector ($\Delta P$) located at the point $P$. With this meaning, the equation above becomes a literally true statement about applying differentiable functions to vectors, and this gives us a fairly direct way to think of vectors as referring to points "infinitesimally close" to the ordinary points.
Sometimes, $\Delta x$ is just an elaborate symbol for another variable that doesn't really have anything to do with $x$. e.g. we might see the formula for the derivative written as
$$ f'(x) := \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$
which doesn't mean anything different from
$$ f'(x) := \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$
Another example is the Taylor series formula
$$ f(x + \Delta x) = \sum_{n=0}^{\infty} f^{(n)}(x) \frac{(\Delta x)^n}{n!} $$
The use of the symbol $\Delta x$ rather than some other symbol is simply to help remind the reader how we are going to use it in formulas.