Another reason Jared Leto's memorable face tattoos are missing in the Snyder Cut of Justice League is that his character provides more of a mental motivation rather than a physical threat. According to Vanity Fair, Snyder's version of the Joker appears in a dreamlike, post-apocalyptic vision to Bruce Wayne (Ben Affleck) as he warns the superhero about what will happen should the band of heroes fail to stop the alien villain Darkseid (Ray Porter). It's this haunting, white-faced version of Leto's character that Snyder envisions Batman, who has a long history with the Joker, carries with him.
Also, the inclusion of the Joker in Justice League wasn't something Snyder had originally planned. As Winter Is Coming notes, Jared Leto's Joker hadn't had a major interaction with Ben Affleck's Batman yet, so it's not a shock that his character has a new, unique look for their first major meeting. The decision to include Leto's character was only made when Snyder had the greenlight from HBO Max to pursue a full vision. That more or less gave him carte blanche to reimagine the character for Leto free from all the Suicide Squad baggage.
Ultimately, why the Joker's face tattoos are absent in Zack Snyder's Justice League is still something of a mystery, as Snyder told Vanity Fair, "I would say that there's been some water under the bridge. Who knows what's happened. I don't know if he's wearing makeup, I don't know what's happening. It's hard to say exactly."
Zack Snyder's Justice League premieres on HBO Max on March 18, at which point we'll hopefully find out if Darkseid sprung for Joker's laser tattoo removal treatments.