If all you could measure was "up" and "down" then I think entangled particles would be indistinguishable from unentangled particles that were created as up/down pairs. But particles can be measured in other directions, and that's where the determinism goes away.

A nice thought experiment is the CHSH game. It's a two player game where the players (player A and player B) cooperate to beat the house. It is played as follows:

1. Each player is assigned a referee.

2. The players, accompanied by their referee, go to separate rooms. Before going to the separate rooms the players can confer. They may also bring any equipment with them that they want. The rooms are shielded to block any communication between the players during their time in the rooms. You may assume that the communication blocking is 100% effective.

3. Each referee uses a true random number generator to generate a bit, and tells the player the value of that bit.

4. The player then generates a bit, by any means, and tells it to the referee.

5. The referee records the bit they generated and the bit provided by the player.

6. Steps #3-5 are repeated 999 more times.

7. After both players have gone through #3-5 1000 times, the referees confer and check their records. For each round the players win $100 in these two cases:

The players generated different bits and the referees both generated 1
The players generated the same bit and at least one referee generated 0

In a classical universe the best strategy for the players is simply to agree on an algorithm that will result in them picking matching bits every round, such as "always pick 0". 75% of the time the referees will generate 00, 01, or 10 and the players will win $100.

In a quantum universe the players can do better. They can generate 1000 pairs of entangled particles and each take one particle from each pair. Let's assume that the particles are linearly polarized photons polarized in the up/down direction.

When player A is given the referee's bit, A sends their particle from the first pair through a polarizing filter and reports a 1 if the particle makes it through the filter, and a 0 if it is blocked.

If the referee's bit was a 0 the player orients their polarizing filter along the up/down axis. If the referee's bit was 1 they orient their filter rotated 45° to the right.

Player B does a similar thing, except their filter is rotated 22.5° to the right if they got a 0 from the referee and 22.5° to the left if they got a 1.

Here's a diagram of their measurement angles, where X0 means player X got a 0 from the referee and X1 means they got a 1:

B1 B0
| | |
| | |
+--------+--------+--------+
|-22.5 |0 |22.5 |45
| | | |
A0 A1

They do this for each round, using the photons from the n'th entangled pair for round n.

Note that if either player receives a 0 from the referee the angle they use will be 22.5° apart from the angle the other player uses no matter what bit the other player got from the referee.

When the measurements on a pair of entangled particles are taken at an angle θ the results match the result you'd get at a 0° difference cos^2(θ) of the time.

For 22.5° that's 85.4% of the time, so when either referee generates a 0 they players will win 85.4% of the time.

If both referees generate 1, B measures at -22.5° and A measures at 45°. That's 67.5° apart and the player bits only match 14.6% of the time, but when both referees generate 1 the players want to generate different bits so that's good. They players win 85.4% of the time in this case.

That's an 85.4% win rate in all cases, which beats the 75% that they can get in a classical universe.

If you try to make some sort of classical-only thing that can take the place of entangled particle pairs you'll find that you can't make it work. You won't get past 75%.

Another thought experiment that might be clearer (or might just muddle things) involving two mysterious devices that you are trying to reverse engineer is here [1]. That puts it more mechanical/computational terms which may be easier to play around with.

[1] https://news.ycombinator.com/item?id=35905284