How does the Velocity Verlet method differ from the standard Euler method? Why do we need to add Acceleration / 2 to calculate position?

The main difference is that it's "symplectic", which is to say it preserves the area of the system. The way you solve these equations you can easily lose elements of the system as a whole while rather accurately simulating the system at the small scale. It's like losing the forest through the trees. If you use Euler's method to simulate a planet traveling around a star, the planet will drift away into outer-space. Because the error in the system will propagate and lose a bit of curve each time causing it to drift away.

This problem is fixed by making sure the equations you use are symplectic in that they preserve the overall area of the space. In different domains this means it keeps the angular momentum of the system better. It ends up being more accurate accordingly. It's used a lot for example Newton's solution to the Kepler's 2nd law wherein he did a straight line to adjust the momentum then simulated the change in velocity to show that both of those operations could not change the area during the same period of time. So at infinite precision nothing could change the area of the sweep (it's sometimes called the Newton–Størmer–Verlet method).

Generally most these methods have reasons behind them but they often lag velocity to after the the change in position, so that you end up properly preserving the energy of the system. Euler's method will start flinging the stuff that should be curving in a circle, a bit too far away as all the error will move further from where it would have curved if we had infinite precision. Using the method we perform the result of the velocity then kick the velocity to where it should be during the half iterations. We still have error on the order dT, but we preserve the much about the rest of the system.

There's a bunch of different ways to do this sort of stuff, so you don't need to use that exact method but if you want to do a bunch of stuff curving around stuff, you should avoid having it all fall apart because you didn't maintain properties of the overall system.

The Verlet velocity method itself uses a special trick to skip the velocity stage. So if it's the method you're using, you need to use half the acceleration because that's what you need to do.