Time integration with implicit Euler is unconditionally stable. This means you can choose arbitrary large time steps without worrying that your solution "explodes". In contrast to explicit Euler, it requires you to solve a linear equation system at least once, which is generally much slower than simple matrix-vector multiplication. If the problem is non-linear, you also need to iterate until your error is minimized. This obviously takes even longer.
However, stability issues can be a pain to deal with and all explicit methods have limitations regarding their stability. Solving stability problems might force you to reduce your time step size drastically and to calculate so many intermediate steps that the implicit version gets more effective. An additional factor is the size of the system you want to simulate. If I remember correctly, the finer your mesh (smaller cells), the smaller are the timesteps you need to maintain stability.
I'd like to know which algorithm uses the least amount of operations per frame / vertex.
I think there is no general answer to the question because it depends on many factors like the mesh size, target FPS and the maximum local velocity that might occur. You need to evaluate all relevant factors and pick a method based on your exact problem.
In case you want raw speed and do not encounter any stability problems, the explicit Euler method should be one of the fastest methods but it usually blows up rather quickly.
A time-integration method that is quite popular in real-time engines is Runge-Kutta 4. It is also an explicit method and doesn't require you to solve a linear system. It has a good balance of stability and speed, so you might have a look at it. I have never simulated cloth myself, nor have I programmed an RK4, so I can't tell if it is a good choice here.