Are they the same in terms of their algorithm? Or do they differ in their respective detection methods?
Haar-like features are, as the name says, features. They are basically some filters, just like the CNN filters. The main difference is that CNN learns the filters by training, while Haar filters are hand designed. These filters are called Haar-like features due to their similarity with Haar wavelets.
Haar cascades are a bunch of Haar-like features arranged into a classifier.