Often when describing entanglement in an informal way, we talk about perfect correlation or anticorrelation of measurements of faraway particles, but I don't see how such correlation is particular to entanglement. Imagine I had a black ball and a white ball, I put them into two separate opaque boxes and I mix them up. One goes to Alice, who stays on Earth, the other goes to Bob, who leaves for the other side of the galaxy.
At some point Alice opens the box and finds the white ball (and she had a 1/2 probability of finding it), so she instantly knows that Bob will find the black ball. Obviously there is no entanglement here, this is just a correlated classical distribution
$$ \frac{1}{2}|bb\rangle\langle bb|+\frac{1}{2}|ww\rangle\langle ww|$$
where $b$ stands for black and $w$ for white. If instead the balls were particles in a maximally entangled state $\frac{1}{\sqrt{2}}(|bb\rangle+|ww\rangle)$, we can think of $b$ as spin up and $w$ as spin down. Then the state would be
$$ \frac{1}{2}|bb\rangle\langle bb|+\frac{1}{2}|ww\rangle\langle ww|+\frac{1}{2}|bb\rangle\langle ww|+\frac{1}{2}|ww\rangle\langle bb| $$
If Alice and Bob repeat the same experiment with these entangled balls, they will observe the same correlation, each time Alice measures $b$, Bob will measure $w$, and vice versa.
What is then the observable difference between these two distributions?