There are a number of threads on this very topic, but in short -- they do different things.
There are crossover features like highlights, saturation, sharpening, but Photoshop takes off in a different direction with layer-based editing/creation.
AP/LR do not function in the same way but are very powerful for editing and managing/archiving large numbers of photos. (Dodge & burn was a big step for Aperture.) With AP/LR, you can sort by any snippet of metadata and batch process, export to a web gallery, etc - from thousands of photos in your library. Photoshop doesn't archive photos in a library.
Some people use one, some people use a combo. Photoshop and AP/LR are different enough that comparison is not easy, or even worthwhile, depending on the usage.