You have two boxes separate from each other. One box contains nothing. The other box has a piece of paper with the number zero on it. The first box represents while the second represents . Two different things. The first has no objects, the second has only one.

Nothing is the most fundamental concept of the three. It typically relates to non-existence or absence of something. However you can actually say a great deal about Nothing as vacuously true statements are always true. Python code uses the None keyword as a token to represent this concept but mathematically it's not useful to do so since we don't have to run it on hardware.

Now the next construction is the empty set. Sets are characterized by their members, which is to say which distinct objects exist within the set and I can recover those elements from the set. However, if no elements in the set exist, then it has Nothing in it. I capitalize to draw attention to the fact we're still using Nothing like a noun here but it reads correctly in natural language as well. Note that it's just acting as the antithesis of existence here.

So the empty set exists. It has the little braces after all $\{\}$ and that's not Nothing. So on the set level we have existence, but on the element level we have non-existence, i.e. Nothing. This works exactly the same way a box does. What's in the box? Could be something, could be Nothing.

Finally the natural numbers are used for counting. I prefer including $0$ as a natural number for this reason and as a number it exists, so it's not Nothing. A simple case would be a set of oranges and (disjoint) set of apples. I can count the oranges and apples then add them together to get the total.

So what if I go on an apple binge and now I only have the empty set. Well, to count those I'd add $0$ to the total. In fact, $0$ is the only natural number such that $x+0=x$ for all $x$. In modern algebra we typically define $0$ this way as the identity element in some system since it generalizes well to more complicated objects. For example the $0$ vector plays a major role in linear algebra.

It's this property that makes $0$ and the only way to count Nothing. In fact, the empty set is an identity element with respect to set union, which is to say $\emptyset \cup X =X$ for all sets $X$. So they do have that commonality among them.

So in short, nothing is Nothing which in turn is a state of non-existence. The empty set exists but contains Nothing. The amount of Nothing we have is $0$. The distinction is subtle but they all clearly have their own role.

The rule that $\mathbb{P}(A) = |A|/|\Omega|$ (for sets $A\subseteq \Omega$) arises in a specific probability context, where you have a sample space $\Omega$ containing a finite number of outcomes all with equal probability. In that particular context, you you are correct that $\mathbb{P}(A)=0$ only when the set $A$ is empty. However, probability theory deals with much more general situations than this, so that rule holds only in a very narrow class of cases.

More generally, probability theory deals with cases where there are events $A$ that are non-empty sets of outcomes, but they still have zero probability. This leads us to make a distinction between events that are certain to occur and events that are merely almost sure (i.e., occur with probability one).