when should I use a power adapter as opposed to a wall adapter

Usually a wall adapter is a power adapter. In other words they are the same thing.

Some refer to the converters that plug into a power adapter to change the prong layout and structure as wall adapters as they adapt a power adapter connector to a specific wall outlet type. The kind you use when you travel, and you need to plug an American power adapter into say a U.K. wall socket.

Outside of that, there is no difference, and real time to use one over the other as they are the same thing.

It's like asking when do you use a car as opposed to an automobile.

The smaller one is the 5 watt charger. It can be used to charge iPhones and iPods, and some iPad Minis yes.

Larger sized iPads require more power, and you should be using the larger one which is either the 10 or 12 watt one.

The iPad came with a power adapter and a cable. You plug the cable into the power adapter, plug that into the wall, and the other end of the cable into the iPad. The adapter that plugs into the wall is called a power adapter. As I asked before, what are you considering a power adapter that is different from what was supplied by Apple?

I'm not sure exactly what you are referring to, however the description that Apple uses for the adapter that you plug into the wall is a power adapter. I don't know what you are using to indicate the differences.

I'm glad you were able to get an answer. Phil0124 provided you with the things I was looking for. So, to address your next question here, does the iPad charge with either of them? Like Phil said, it has to do with the amount of power. If the iPad will not charge with either of them, do you have anything else to try charging, like an iPhone? Have you checked the cable? Have you looking in the charging port to see if there is any dirt, lint, or other debris in the port that could block the cable being plugged in correctly? See this support document for other charging suggestions. If your iPhone, iPad, or iPod touch won't charge - Apple Support

Would you please post a photo of these two adapters. I am just to confused with what you are talking about.

What exactly are you using to charge it?

Does it look like this:

or like this:

See here: About Apple USB power adapters - Apple Support