Chaos is unpredictable
Mankind seeks to understand how the world works, what rules govern our existence. This is the goal of all science.
The reason we want to find these rules is that they allow us to make predictions about what will happen in the future when we take certain actions.
The rule — or in science parlance: the theory of gravity states that if I release an unsupported object, that object will accelerate towards the ground at 9.8 m/s2.
This is useful to know... it allows me to know to avoid performing that trick with my own body, because my body does not take well to falling for too long and then experiencing the sudden stop that happens when it reaches the ground.
Now how much is "falling too long"? If we were to look at gravity for what it really is — a contiguous field around every particle of mass that is warping spacetime(*) — we would end up with some pretty complex maths that is really hard to calculate.
But we do not have to use that, because we can take a shortcut and use a much more simple formula. Say we want to avoid impacting the ground at a speed more than 10 meter per second... we can calculate the maximum allowable falling height by the simple formula. h = 102 / (2 x 9.8) = 2.5 meters.
In other words: Gravity is predictable, and we do not have to actually take a fall to see what would happen if we did.
Chaos is when you cannot take any shortcuts
Chaos on the other hand is where you cannot calculate beforehand what will happen. You have to actually go through all the steps and let things run their course to see what happens.
Some 20-sided dice
These behave chaotically
Put these in a cup, cover the cup by your hand, give them a really good shake, and pour these dice out on the table. What will the result be?
In theory this could be calculated. If you knew the exact material composition of the dice, the cup, the hand that covers it; their starting positions and initial velocities; and if you knew exactly how the shakes will go, you could calculate what the result will be.
But the problem is that there are way too many variables involved, and even the tiniest variation of the starting-conditions of the variables or the shakes can affect the outcome.
So even though the rules are well known to you, you cannot make the prediction. The process of rolling dice is chaotic.
All randomness is chaotic, not all chaos is random
You might say "Ah, but throwing dice is a random, non-deterministic process, I was talking about mathematical sequences that are not random, but entirely deterministic". Well, this is true, but it does not make them any less chaotic. They are pseudorandom. And while you can recreate any deterministic sequence, with some sequences it is not until you have actually created the sequence — at least once — that you know what you will end up with. You cannot take a shortcut and predict what you will end up with.
But what is we could predict everything?
Suppose we know everything... we know the position and momentum of every single particle in the known universe. And we know thew laws that govern them. And suppose we put all that knowledge in a simulation, ran the simulation, speeding up the time. Surely then we could predict everything that will happen?
Well... no. Because the simulation itself would have to be part of that simulation, which leads to an infinite regress that makes it impossible. This is related to the Halting Problem, which stems from Gödel's Incompleteness Theorem.
Hence there will always be things that are unpredictable. And when you have unpredictability, you have chaos.
So yes: chaos exists, at least for us puny humans.
(*) No, I am not making that up with Star Trek technobabble, this is what gravity is