Even though this method is more widely used than the simple Verlet method mentioned above, it unfortunately has an error term of O(Δt^2) , which is two orders of magnitude worse. That said, if you want to have a simulation with many objects that depend on one another --- like a gravity simulation --- the Velocity Verlet algorithm is a handy choice;
I found that here and the site doesn't offer any explanation on why this would be the case. Apparently the advantage over the Verlet, that the velocity Verlet, has to offer is that it can calculate the velocity and the position but I don't see how that helps. I haven't found any other sources that help either.