Heroic is better than Lutris for Epic, GOG and Amazon because it doesn't need their respective launchers or installers. It basically downloads just the game files without any intermediary archive, installer or launcher. So game installation is much faster. Heroic also supports game updates and cloud saves.

Heroic accomplishes that by using open source command line clients for the respective shops. I really wish Lutris would do the same. Would make it much better.

What makes Lutris better is that it supports even more stores and launchers and even emulators. It also has Steam integration. So it can contain all your games. Unfortunately it does not display all your available games in one combined library view, which makes finding uninstalled games much harder. But supposedly they are working on getting a combined view. No idea what's taking them so long.

Lutris used to give you more tools to configure Wine and other helpers, but Heroic is quickly catching up.

For me both have their use cases. Heroic is my go to for GOG and Epic games and everything else not Steam (e. g. old Windows install files that I still have lying around). EA App only ran for me on Lutris until one day it didn't. Then I tried with Bottles and it worked ever since. Not trying to add more confusion, but for some people Bottles has a more intuitive interface than Lutris.

Heroic can support EA games and other launchers by basically doing the same thing Lutris does. You just install the launcher with Proton and use that.