There shouldn’t be a difference in both ways of starting, as ultimately both execute a lutris launch command. What do you want to check I don’t get it. I don’t have it installed anymore.
The question was if is there or not difference between those two ways to open battle.net.
You already answer indicating “…shouldn’t be a difference”, the words indicates more an estimation than a confirmation, but once you also already deleted the app, there is no reason to install it back to check.
One advantage of opening Lutris is the fact that it makes easy to stop the process if battle.net didn’t launch properly, while not opening Lutris we need to kill process using command line o live with multiple process.
For a while I can live with this issue.