I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..
I can't find myself in all of this. Why there are so many infinities, and why even bother to classify infinity, when infinity is just... infinity? Why do we use all of these symbols? What does even each type of infinity mean?
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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.
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The major motivation for classifying infinities (other then the intrinsic enjoyment of mathematics) is that different infinities permit different properties.
Countable infinities can be reasoned about using inductive proofs. On the other hand many of the properties analysis makes use of requires uncountable sets.
Having different infinities often makes showing that two sets are not isomorphic by allowing us to determining the cardinality of both.
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If it's any consolation, in practice you won't encounter more than two different types of infinity: that corresponding to the natural numbers, and that corresponding to the real numbers (cardinality of the continuum).
The reason why it's necessary to differentiate between those two is that the "size" (cardinality) of the real numbers corresponds to the cardinality of the power set of the natural numbers.
An elementary theorem states that the "size" (cardinality) of any set is always strictly smaller than that of its power set, and the same holds true for the natural numbers.
Hence many of the tricks that work for the "infinity" associated with the natural numbers do not work for the "infinity" associated with their power set/the real numbers.
For example, you can "add up" some sequences of "infinitely many" non-zero numbers and still get a finite result, provided that the infinity involved corresponds to that of the natural numbers.
However, if the infinity involved corresponds to the real numbers (i.e. an uncountable sum of non-zero numbers), the sum will always be infinite, no matter how small you try to make the individual terms.
It is discrepancies like these which necessitate that we differentiate (in particular) between these two different types of infinities. However, at least speaking as someone who specializes in probability and statistics, any other type of infinity does not occur as often.
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Your question touches on issues both mathematical and philosophical; in fact, you might enjoy an introductory text in philosophy of mathematics and logic, or on the foundations of mathematics.
Truth is, an infinity is not just an infinity. One reason for this peculiarity is due to the seminal work of Georg Cantor.
Cantor proved that the set of real numbers is greater than the set of natural numbers.
Crazy.
Even crazier is how he used grade-school concepts, such as a one-to-one correspondence, to do so.
But none of this really answers Why?. Maybe I can give you something of an answer. Mathematics is a language rigorously defined. It's one thing to throw around a natural-language word like infinity, and totally different to empower that word with a thoroughly defined meaning in a formal language.
Put more poetically, Cantor provided a definition of infinity by showing that infinite sets of things can be classed and organized like books on a shelf.
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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).
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One natural use for smallish infinite ordinals, which makes clear that different infinite ordinals have distinct properties, is the length of games.
I'm playing a chess-like (turn-taking, deterministic, complete information) board game of some kind against an opponent. My teammate is also playing a separate game against my opponent's teammate. Moves are made in both games at the same time.
Sadly, I'm losing my game, and can no longer do anything to avoid an eventual loss. Luckily, my teammate is winning, and the rules are such that as long as I can delay my loss for long enough for my teammate to force their win, we're OK.
Borrowing chess terminology, my opponent's position can be described as win in $n$ inductively, by saying that they win in $0$ if they've already won, and they win in $k+1$ if no matter my next move, they will be able to guarantee ending up in a win-in-$k$ position.
I'm losing, so my opponent has win-in-$p$ for some $p$. My teammate has win-in-$q$ for some $q$. I'm OK exactly if $p > q$ (what happens if $p = q$ isn't important for what follows).
So far, it seems like we can measure the length of winning sequences using natural numbers. But then I spot something interesting: I actually have infinitely many moves available to me, and I can see a move that leaves me $1$ step from defeat, one which loses in $2$, one which loses in $3$, and so on. Every move seems to lose ultimately, each in a fixed finite number of steps, but I can at least choose how long it takes for me to lose. So if my teammate is in a win-in-$k$ position, I just need to pick the move that loses in $k+1$ moves and I'm all right!
In some sense, then, the winning sequence (losing sequence) that I'm facing is "longer" than any finite sequence my teammate might need, despite the fact that in all circumstances I lose in a finite number of moves. For this particular situation, we might say I will lose in $\omega$ moves, where $\omega$ is some kind of special number (spoiler: ordinal) that is bigger than any natural number.
However, I notice my teammate is actually in the same exact situation: they will win, but their opponent will be able to choose, on their next move, how long it takes. Once they've made that choice, their fate is sealed, but if they make their choice at the same time as I make mine, I can't ensure that they won't pick a longer finishing sequence than me. So I need to find a move such that I can put off my decision until my next turn, see what my teammate's opponent picks, and just pick a longer losing sequence. If you remember our inductive definition of win-in-$k$ from before, you might say that I'm looking for a lose-in-$(\omega + 1)$ situation, i.e. for a way to ensure that I'm in a lose-in-$\omega$ situation next turn.
Once you can imagine that, you can perhaps imagine losing in $\omega + 7$ moves, or even $\omega + \omega$, $\omega + \omega + \omega$, even $\omega^2$ and things more exotic than that. You can come up with a natural comparison ordering of these game lengths that precisely corresponds to whether or not I'll be able to guarantee that my teammate wins before I lose. If you do that, you have a game length for every natural number, then you have a game length which is larger than any natural number, then you have many more, still larger, game lengths. We can consider each of these long game lengths as a variety of infinite numbers that are each distinct and comparable with each other in meaningful ways, and as you might've guessed there's a meaningful addition (or at least concatenation) concept on these lengths which satisfies some (but not all) of the properties of addition on natural numbers. These are (some of) the ordinal numbers.
All this to say, there isn't simply one infinity that sits at the top of all numbers. There are many objects that don't seem to have this property or characteristic of "finiteness", indeed as rich and diverse a variety of non-finite objects as there are finite ones.
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 $\aleph_1$.