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.
