Many sources present the Euler, Verlet, velocity Verlet, and leapfrog algorithms for integrating Newton's equations. Based on the order of accuracy, it is agreed that velocity Verlet, Verlet, and leapfrog are superior to the Euler method. It can also be shown that the Verlet and velocity methods are algebraically equivalent. What are the advantages and disadvantages of using velocity Verlet or leapfrog integrators? Is one more accurate or expensive than the other?
The shortest answer is that the Verlet / Leapfrog methods are symplectic and time reversible, and these are desirable numerical properties that reflect the physical reality of certain simulation problems.
Sympletic means that when the methods are guaranteed to conserve total energy (more correctly, the Hamiltonian) in conservative simulation problems. Take for instance, the simulation of the earth spinning around the sun. Without an external dissipation of energy, the earth is guaranteed to remain indefinitely in orbit. But using Euler or RK4 to perform the simulation, for example, the earth would eventually leave orbit or crash into the sun, due to a small numerical drift in the total energy that builds up over a long period of simulation. Symplectic methods like Verlet / Leakfrog explicitly preserve the total energy, so the numerical solution is also guaranteed to keep the earth indefinitely in orbit.
Time-reversibility means that the (fixed-step) method can take $k$ steps forward in time, followed by $k$ steps backwards in time, and arrive at the same initial conditions used to start the simulation. We often apply time-reversible numerical methods to time-reversible physical processes; we then expect the numerical solution to have long-time behaviour similar to that of the exact solution.
But these two properties come at a sacrifice of accuracy and stability. The RK2 method is a comparable second-order method that is significantly more accurate, and the BDF2 is a second-order method that is significantly more stable. Also, these methods are explicit (i.e. efficient, because they can be computed without solving a nonlinear system of equations) only for certain physical processes; their most general form is implicit.
- Hairer, Lubich, Wanner, "Geometric Numerical Integration", Springer 2006
EDIT v2: the Leapfrog and Verlet methods are (theoretically) identical. The following quote is from Pg. 7 of Hairer Lubich Wanner.
This basic method, or its equivalent formulation given below, is called the Stormer method in astronomy, the Verlet method in molecular dynamics, the leap-frog method in the context of partial differential equations, and it has further names in other areas.