The key to understanding the differences lies in how car detailing and paint restoration works.
The key word here is aggression, referring to the "rate of cut" of the pad.
Here are the pads rated from most aggressive to least aggressive based on the information provided by 3M.
1. Compounding pad
Compounding refers to the process of "levelling" defects in the clearcoat (or paint in some cases) with chemical abrasives commonly referred to as "cutting compounds".
The severity of the paint defects (swirls, scratches, etc.) determines the degree of aggression required. The idea behind detailing is to take off the minimum amount of clearcoat/paint required to remove the defects.
Compounding pads are the go-to pads for relatively aggressive cutting action. They will usually have a stiffer foam to them, and the website also indicates that they have the most aggressive rate of cut.
2. Buffing pad
You'll see that the 3M site considers the buffing pad to still be a compound pad, just less aggressive. Ideally it is used with less aggressive cutting compounds
3. Polishing pad
Polishing compound is less abrasive than cutting compounds, so the degree of cut required here is far less than when compounding.
4. Finishing pad
This is the least aggressive of all pads. It is the one that is closest to a wool pad, only less messy and less likely to impart scratches on the paint work. This one is normally used in conjunction with a finishing compound which has much finer grit in it to make the polish "pop" and stand out even more.
If one wants to add a wax (like carnauba wax) on top of the paint to using a machine, this would be the go-to pad due to its minimal abrasion.
One might ask...
... Whether it is possible to mix and match pads with compounds/polishes, so for example use a buffing pad with more aggressive compound or compounding pad with less aggressive compound.
The answer is yes. It is usually done to provide an intermediate level of cut where Pad A/Compound A is too aggressive and Pad B/Compound B is not cutting enough.
There's a very nice YouTube video by AmmoNYC which explains this in depth.