The difference between "store up" and "store away" lies in their actions. "Store up" means to gather or collect items over time for future use, like saving canned food for emergencies. On the other hand, "store away" involves putting something in its proper place to keep things organized, such as neatly putting books on a shelf after finishing homework.

"Store up" typically means accumulating or collecting items over time, while "store away" refers to putting things in a designated place for storage. So, you might "store up" supplies for a project, and then "store away" those supplies in a specific location.

A small difference between American and British English in British English the two things mean the same as already described by others. But in The U.K you will often also hear "stock up" to mean store up and sometimes you might hear "stow away" to mean store away. this "stow away" likely derives from the British Nautical history and is something you would have heard in the Navy and armed services.

The former emphasizes accumulation of something. "Up" creates the image of a pile. As you build a pile, you have more and more of it. The latter emphasizes the placement of something. "Away" creates an image of movement. The further away something goes, the less you become aware of it.