The Wrestling Ampersand
The 'meme-' root/word is not synonymous with 'mime-'. The words are related, in that memetics works via mimicry - a meme propagates upon human mimetic iteration. The human is the information carrier/copier.
The distinction between the two words, 'meme-' and 'mime-', is where memes also move via artifacts. You don't really do anything with a book, or a painting. So when you use one of these artifacts, and another person watches, and goes to mimic you, they may not truly 'see the picture'.
Mimicry is not the exclusive means of memetic propagation, and that is where the two words differ.
Also, mimesis is a word, but memesis is not. Unless one is referring to the genesis of a meme, which could easily be called its memesis, but memesis isn't being talked about these days anyway. Or at least when its is, it's equated and conflated with 'mimicry', like man wrestling with the ampersand.