The first thing to know is that there are two completely different kinds of modes and they work very differently from each other. One kind is for your home automation rules and the other kind is for your security rules.
We had the ones for home automation first, so a lot of the documentation doesn’t even discuss the ones for security. That was true for this FAQ at the time that it was written.
Location modes are used for home automation rules
Location Modes are definitely confusing. And weird. But useful.
The first thing to know is that there is no functional difference between them. They are just arbitrary values that you yourself define. You could call them grape, banana, and orange, and they would work the same way.
They don’t have any real meaning until you start using them in automations.
The easiest way to explain them is through examples. The following examples are all from the classic app, not the new app, but they should give you the concept.
Security modes represent the armed state for smart home monitor
If you are asking about the “security modes” that are used with smart home monitor then those have a very different structure. They are pre-defined as away, armed stay, and home. You can’t add more and you can’t change how they are used.
The easiest way to understand those is the following thread (the topic title is a clickable link). You will have already read the routines FAQ that it links to, so you don’t have to go around that particular circle again. But it will discuss how the security modes are used and how they differ with the ADT hub model.
I’m sorry if all of this seems like a long way around, it is, and it has to do with the fact that the system has grown over time and then they gave some new features the same name as some old features. 
(If somebody wants to write a new modes FAQ for the new app, that would be very welcome. I can’t do it because the new app is not voice accessible.
