If you're asking what's the difference between similar Austins and Morrises, it depends on the model.
In the case of the Mini and Mini-Cooper variants, not much - truly badge engineering only. If you take the grill and badges off, you'd have to be a real nerd to say which was which.
If you're asking how Mini Coopers differ from Minis, I'd advise you to use the interent to find articles. The Wikipedia entries aren't bad, though I'm not usually a fan.
Going back to the Austin vs Morris differences, it's generally true to say that Austins were usually the baser models, Morris a little more upmarket, then Riley and Wolseley, with MG being the sporting variants. That was pretty much the case for BMC in the UK, but I know they used different badges on export models so that may not have been true over here (USA).