How is an infinitesimal $dx$ different from $\Delta x\,$? When I learned calc, I was always taught 
$$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$
But I have heard $dx$ is called an infinitesimal and I don't know what this means. In particular, I gather the validity of treating a ratio of differentials is a subtle issue and I'm not sure I get it. 
Can someone explain the difference between $dx$ and $\Delta x$? 
EDIT: 
Here is a related thread: 
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
I read that and this is what I don't understand:

There is a way of getting around the logical difficulties with infinitesimals; this is called nonstandard analysis. It's pretty difficult to explain how one sets it up, but you can think of it as creating two classes of real numbers: the ones you are familiar with, that satisfy things like the Archimedean Property, the Supremum Property, and so on, and then you add another, separate class of real numbers that includes infinitesimals and a bunch of other things.

Can someone explain what specifically are these two classes of real numbers and how they are different?

The term "infinitesimal" was used by Leibniz. This was at a time before the concept of limits, as we know it today. He still thought of $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ as a quotient with $\mathrm{d}y$ & $\mathrm{d}x$ being very small.
Today $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ is not a quotient but is notation for the limit after the limit has been applied, i.e. the whole thing is notation for the derivative.
Another way of looking at your formula is

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

The current notation can be very misleading
