Heroic is way easier to use imo, but both are good options https://heroicgameslauncher.com/
1. it doesn't (at least recently?) always do a great job of handling multiple displays, either launching games on my second monitor, which I orient vertically or getting confused about which monitor to use and switching back and forth until eventually the instance (but not the Lutris client) crashes
2. I find myself getting into launcher hell where I'll use a different wine version for one game and when I switch to a different game, it's using this new wine version and stops working
Not sure if Heroic solves these issues but I would try it again (didn't have any luck setting it up initially) if it does
If you are using KDE then there is a global Window rules setting and Heroic actually obey those rules, so you can force to launch a game always on X display, always minimized etc.
https://docs.kde.org/stable5/en/kwin/kcontrol/windowspecific... (not sure that's the most up to date manual)
It looks like gamescope has it's own fullscreen shortcut (super + f), but people complain about it about it going to the wrong display. Maybe your window manager offers a more consistent full screen option like i3 does.