Consider the earth orbiting around the sun. We know the interaction and can solve for the orbit.

From the force you calculate the earth’s orbit and with initial condition you know where it is at all times.

Given the Hamitonian H one can calculate the quantum wave function that is equivalent to the full QS.

Know how the atom will behave in terms of the wave function.

Suppose the problem is too difficult to solve because the interaction is more complicated.

For the two slit experiment you propagate a particle to slit 1 (the interaction) and then to the screen.

For this I can write down an amplitude to reach the screen by nature of the interaction.

One difference in QM is that I need to find all possible ways to reach the ending state

I can imagine a more complex system with a series of interactions.

My goal would be to calculate the amplitude to travel along any given path and then sum over all possible paths.  Feynman diagrams diagrammatically represent this deeply complicated mathematical process.  A first order diagram in some sense shows the interaction as a single step process, as if you can get a good result by correcting the trajectory one time classically. High order diagrams are similar to the multiple corrections required to find the asteroids final trajectory.  In addition to the expansion of the interaction Feynman diagrams represent the multitude of ways that quantum systems can travel. The amplitudes for each trajectory will be included and interference may result.