Why do we classify infinities in so many symbols and ideas?

As an insight, think of the size of the usual sets of numbers.

Think of the set of all positive integers, and the set of all positive multiples of 5. It's a bit strange to the "uninitiated", but it's ultimately not hard to wrap your head around the fact that there's the same amount of each, because we can list them side by side without any problem:

$$\begin{align*}1 &\to 5\\2&\to10\\3&\to15\\&\dots \end{align*}$$

Each is an infinite set, and while you may be tempted to think that there's five times more numbers in set of positive integers than the set of multiples of 5, I just showed above that you can make a 1-to-1 correspondence between them, so in fact they are the same size. I'm just making up this notation, but we can call this $\infty_1$.

Now think about the set of all numbers between $0$ and $1$. There's no possible way, no matter how you try, to get a 1-to-1 correspondence with the set of positive integers like we did above. We can try:

$$\begin{align*}1 &\to 0\\2&\to0.1\\3&\to0.01\\&\dots \end{align*}$$

But what about all the numbers between $0$ and $0.1$ that we're missing? We get this rough intuition that there's more numbers in the second set this time than in the first. We can call this $\infty_2$.

Already we have described two "different infinities", just by looking at a couple sets of numbers. $\infty_1$ is countable, as we saw in the first enumeration, but $\infty_2$ is not.

To develop the entire concept of aleph numbers and transfinite sets and etc requires some pure mathematics that is past my pay grade. But maybe you can see how infinities can be classified as different.

In the mathematical world, $\infty_1=\aleph_0$ and $\infty_2=2^{\aleph_0}$, so you'll see it written that way. As @Milo Brandt pointed out, for those who are more interested in the rigor of transfinite sets, etc, keep in mind that $2^{\aleph_0}$ is not necessarily $\aleph_1$.

I won't comment on your more philosophical questions, but I will give what I think is one of the more important applications of different sizes of infinity.

There is a rigorous mathematical way of thinking about a computer program, called a Turing machine. One can show that the cardinality of the set of Turing machines is $\aleph_0$, however the set of all possible problems you might want a computer program to solve is strictly bigger (cardinality of $\mathbb{R}$). The very real application in this case is the conclusion that there are some problems which are not solvable by any computer program.

One issue not yet addressed is why we have both $\aleph$s and $\omega$s. These both exist because Cantor introduced two separate concepts about infinite sets - their sizes ($\aleph$) and their order-types ($\omega$). Everyone else explained sizes, so I won't go over that.

For order-types, begin by considering the following set:

{ 1/2, 2/3, 3/4, ..., N/N+1, ... }

This is of course an infinite set of size $\aleph_0$. But the elements can also be ordered by <, and so this is also a set of a particular infinite order-type, which Cantor chose to name $\omega$. Now consider a variant of this set:

{ 1/2, 2/3, 3/4, ..., N/N+1, ..., 1 }

This is also of size $\aleph_0$ (I leave proof to you). But it is a different order-type, because its internal ordering is structurally different - i.e it is NOT possible to biject the two sets preserving order. This can be seen by noting that the element 1 in the second set has infinitely many predecessors, which no element in the first set has. Cantor names this set's order-type $\omega+1$. If I added 2 into the second set, the resutling order-type would then be $\omega+2$; and so on. Actually, that doesn't do it justice - there are an infinity of 'and so on's coming - compounded 'to infinity and beyond'.

So what Cantor discovered is a very sophisticated notation for describing highly complicated sets of real numbers (or as he thought of it, points on the number line), a notation which he realized could be abstracted to a system of actually infinite numbers having its own rules of arithmetic (inferred from the point sets resulting from catenating other point sets).