Honestly, from a day to day standpoint, by my experience of using both, there's little practical difference between, for example, yay, and paru — it mostly just ends up coming down to subjective, nitpicky meta things about the program itself.

Up until this post, I hadn't heard of Aura, but, after briefly looking at its repo, it appears that it's effectively the same as yay and paru [1.2]; what it tries to do differently is it tries to ensure that there are translations of it (I'm guessing its output) in other languages [1.1.1]. One thing that I'm knee-jerk not super fond of is that it utilizes its own centralized metadata server [1.1.2], though I admit that I haven't thought about that a great deal, so perhaps there are some aspects that about it that I'm missing, or perhaps misunderstanding, or perhaps there's a different way to view it.