It's about time and distance. Out implies shorter time and shorter distance, away implies longer time and longer distance.

You use "out" when you've gone out for the day, or gone out shopping etc, "away" is used for things like away on holiday or away for a few days.

just so you know both of your examples are erroneous. 1. I'll be out next days. this could be corrected to I'll be out for the next few days. 2. I'll be away next days. This could be corrected to I'll be away for a few days. Sorry, i don't know why they're wrong, just that they're wrong. good luck with it all!

I agree. I would say "I'll be away for the next few days" if I was not going to return home for a few days. but for example... if I was going to a friends house and would be back by in the afternoon I would say, "I'm going to be out for the day."