Before highlighting the points of commonality between today's twin tales, its good to distinguish between the two in the areas where they diverge. Specifically, our Earth-One tale had as its alien antagonist the Marshal, genetically superior warmonger of Mars II where J'Onn J'Onzz surviving brethren ended up after Mars proved inhabitable. However, Earth-Two's Atom faced the Marshal's counterpart not on his own world but on his universe's Mars II, wherein the evil alien Butor had attacked the peaceloving Martians. That was a separate tale, not connected with this Twice-Told Tale.
The date was July of 1984, one year before the Crisis on Infinite Earths would threaten the entire Multiverse. Twin forces attacked parallel planets, with the aforementions New Martians laying claim to Earth-One while the Commander of another cosmos invaded Earth-Two's airspace. While each antagonistic alien army dueled with members of the Justice League and Justice Society, they specifically went for the centers of power as they perceived them. For the Marshal, this was by approaching the United Nations with an ultimatium, backed with an arsenal of spacecraft at his disposal. For the Commander, it was by sending his powerful grunts against the United States' Pentagon, consisting of winged apes then legendary menaces.
While the Society battled back and defeated the Commander and his cronies with the aid of Earth-One's Superman, Wonder Woman, Flash and Supergirl, on the quartet's homeworld their League allies barely stymied the incursion of the Martians while losing their prized satallite headquarters. Hence, the Justice League disbanded, in part due to Aquaman's claim that not all members were present to defend the Earth. However, that was because three of their own members were lending a hand aiding the Society, who only had four of their own to tackle their cosmic crisis. So ironically, more Leaguers saving more worlds led to their disbanding, whereas only a handful of Societialites present to protect just their own planet... no problem!
However, there is much left to be told about the Commander who nearly claimed Earth-Two and its entire dimension has his latest prize. Would anyone equal his infamy on Earth-One? Yes, about a year earlier, whom we'll learn more about soon...