Mix Animation
In this tab you will find the settings that allow you to combine conventional key animations - e.g., of Deformers - with a simulation. This is to define via a Vertex map (to be placed in Map) which object points should follow the animation and which should follow the simulation. With this, simulated secondary movements can be created: Imagine a kangaroo with floppy ears. Animate the kangaroo in the traditional manner and only apply a simulation to the floppy ears. If the kangaroo now hops through the outback the ears will also bounce realistically.
Since these settings are found in both the Cloth tag and the Rope tag, don't be surprised if the examples here were created using only the Cloth tag; the functionality for rope tags is the same.
In this example you can see 2 Bend deformer animations. Cloth simulations only affect the crease area. The restriction is made by Vertex Maps. Outside the kink area, the mesh strictly follows the deformer animation.
With Pins method
This method is similar to the clothing belt or rope belt functionality: imagine an adjustable, more or less rigid connection to the animation points for each object point in the simulation (for the tags mentioned, these are visible as yellow lines).
The following settings can be used to set for each individual object point whether and to what extent the animation should influence the simulation.
With Pins is intended primarily for the character animation area.
Activate or deactivate the With Pins method here.
Specify here how much influence the animation should have on the simulation. Values of 0% and 100% will completely simulate or animate, respectively. Intermediate values will produce a corresponding mix.
To set the influence at the point level, drag a vertex map here. Rule of thumb for Influence = 100%: red follows the animation; yellow follows the simulation; intermediate values correspondingly.
This can be used to convert the Vertex map values. If set to 100%, the Vertex map values will be applied 1:1. If set to 50%, a vertex map value of 80% will be applied at 40%.
This is quite practical if you paint 100% values with the 
Paint Tool (so you can see the effect areas directly very well), but these 100% should not be evaluated directly as full influence for adjustment reasons. In other words: you can fine-tune the effect here.
With Force method
This method also compares between the positions of animation points and simulation points, and depending on the Strength set, a corresponding force acts on the respective simulated point in the direction of the animated point.
Since forces (and not positions as in With Pins) are used here, there is no guarantee that the simulated points will ever actually reach 100% of the animated points in their positions. A Vertex map can be used here as well to control the force at point level.
This method is not so good for character animation, it is more for artistic effects.
If you remember Bullet Dynamics, this had similar functionality(Follow Position/Rotation) to mix animation and simulation.
Activate or deactivate the With Force method here.
Use this to regulate the force with which simulated points are dragged to their animated position. If set to 0% only the simulation will have an effect and an increasing number of force points draw it correspondingly more in the direction of their animated positions.
To set the influence at the point level, drag a vertex map here. Rule of thumb for Influence = 100%: red follows the animation; yellow follows the simulation; intermediate values correspondingly.
See Map Scale under With Pins method
In short, this can be used to fine-tune the influence.
Follow shape method
This method is somewhat different from the other two, since it "only" takes into account an animated shape change due to e.g., Deformers, PLA animations or Pose Morph. PSR object animations will be ignored. This means that the object will be simulated as a whole (e.g., falls downwards) but will still follow the change in shape. The object point and polygon count may not change during this process.
Internally, this works by continuously adapting constraint target lengths and angles to the change in shape. This makes it possible to simulate the change in shape dynamically in consideration of collisions and all other constraints.
For this example the spikes point outwards during the simulation. The Soft Body reacts correspondingly to gravity and collision.
Activates or deactivates the Follow Shape method.