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.