A pile is made by putting things, or material on top of other things/other material. Examples are a pile of clothes, or a pile of dirt. You would typically see a pile of clothes in a teenager's room.
A bale is an unordered bundle of things that has been tied up with string or ties, or possibly wrapped up. Examples would be bales of hay, or bales of rags. On a farm, machines take loose hay, compress the hay into bundles, then trim them, and tie them up into bales. Sometimes they would be piled into haystacks, or here, large bales are often wrapped up.