In case your box was classical, it'd output either vertically polarized A and horizontally polarized B or vise-versa, a 45 degree polarizer placed in front of both would allow both to pass half of the time, and block them half of the time - but the pass\block statistics would not be correlated, while in the entangled case, if A passed, B will be blocked, and vise-versa. This extra correlation in the entangled case is what violates Bell's inequality theorem.
A nice use of entanglement is given in the Quantum eraser experiment.
Lets look at the following setup:
photon A goes to a detector, while photon B encounters a screen with two holes before hitting a screen - a hit on the screen behind the two holes is registered only if it coincides with a detection in detector A (so that we know the detector in B actually detected the entangled photon, and it wasn't some noise in the system)
In this setup, we would observe an interference pattern on the screen behing the two holes.
So far, nothing special.
Now change the experiment a bit:
Add $\frac{ \lambda}{4}$ and $\frac{3 \lambda}{4}$ plates on the holes of the screen, so if B goes through one hole it's polarization would be clockwise, and through the other hole it's polarization would be anti-clockwise, and a polarizer which passes only only, say, clockwise polarized light.
Now no interference pattern would be observed.
This is also in agreement with the classical laws of optics.
Now the quantum "wierd" part: Add a linear polarizer in front of A's detector in a 45 degrees to the horizontal (or vertical) axis. The interference pattern will return.
This happens since the measurement in the new basis changes the state of the systems so that it is now in a superposition of vertical and horizontal polarizations, and so the plates in the holes would affect B differently - they would produce a superposition of clockwise and anticlockwise coming from both holes, and would make it possible for them to interfere.
You can look at it this way: There is always an interference pattern, but before the polarizer was inserted in front of detector A, there were two cancelling interference patterns, and so none was observed - but the 45 degrees polarizer filtered out one of these interferences and made it possible to see the other.
Some notes:
The entangled pair in this setup is assumed to be in a polarization basis with the axes horizontal or and vertical.
Also, a bit different (but not much) experiment is nicely detailed in this article: Quantum erasure with causally disconnected choice Pages 3 and 4 contains a nice and more rigorous explanation than the one given here (although it might be a bit much for an undergrad, which is why I did not write it down here).
An even more comprehensive coverage of entanglement can be found here - it's a very good read, highly recommended.
