Greasemonkey existed before even Chrome browser was born and it works only in Firefox. On the other hand, Tampermonky was created later, its user scripts are compatible with Greasemonkey and supports both browsers.
In Firefox, you are free to use either of the extensions and you should get the same results without any functional difference. But in recent ~year, and mostly after the incident you linked to, I started recommending using Tampermonkey. Greasemonkey got more limited and even though this script now works fine now and will probably work some more years, it has very bad documentation and reoslving issues with that was more like a roulette.