From a machine learning pespective, QE is a sequence labeling problem (i.e., whose output sequence has a fixed length and a small number of labels), while APE is a sequence-to-sequence problem (where the output is of variable length and spans a large vocabulary).
we can regard APE-based QE as a "projection" of a more complex and fine-grained output (APE) into a simpler output space (QE).