This is the dual of Use of Reduced Homology. The answers to that question and the fact that basically every homological argument I have seen so far gets easier when using reduced homology left me with the question: Why do we introduce non-reduced homology in the first place?
One argument seems to be that (say singular) non-reduced homology is slightly easier to define, because you don’t need the augmentation map. However, if I’m not mistaken, you can also get reduced homology by defining the empty set to be a $(-1)$-simplex (and it is a face of every simplex in precisely one way) and then proceeding in the same way as you do for normal homology. This seem hardly more difficult.
Is there another reason to prefer non-reduced homology?