The primary difference between Exiles Among the Stars and Worlds on Fire is that this takes place around 18-20 years earlier. EAtS was centered on the Fleet in Exile, which took CASPR's Space Fold Engine to evacuate the Sol system when the Invader's victory seemed inevitable. As such, the Earthers know much less about the Invaders than they did in EAtS.
Mech development with the CASE was been faster than was implied with the old Tactical Standing Frame lore. There had never been an opportunity for Human vs Human mech conflict in the EAtS timeline, since their development was a direct response to the Invaders after they'd reached Earth. CASE (well, proto-CASE at any rate), date back to the Resource Wars, a period of desperate conflict as Earth's ability to support life was dwindling. Seeking to escape the devastation, factions of humanity fled into orbital colonies, or subterranean arcologies, but neither could support the vast population of Earth, so the Surfacers continued killing each other for what resources were left.
This slow, self destructive arc was halted prematurely with the discovery of the Lunar Cradle by spacers, and the promise of a revitalized Earth.
CASPR was founded, primarily by Orbitals, to ensure that the Lunar Cradle's tech wouldn't fall into the hands of Surfacer Warlords, and as the faction with the keys to paradise they managed to install themselves as the de facto government of what would become the Sol Planetary Republic (a representative republic that would eventually consist of Terrans, Martians, Jovians, and Orbitals). CASPR's Regulators would use CASE upgraded Lunar Cradle tech in police actions to keep those who didn't agree with their vision for the future in check -- this included the dismantling of the Mega Corporations who largely owned the subterranean arcologies and maintained a significant population of indentured Vaulters (who had signed restrictive contracts for the privilege of being able to live out the Resource Wars in relative safety and plenty).
Because weaponized CASE were already a thing, instead of rolling over Mars almost as easily as they did the Saturn colonies, the Invaders got bogged down on the red planet. For better or worse, CASPR was managed mostly by scientists and bureaucrats, not soldiers. When they saw the Martians slowly being ground down, they didn't leap to the aid of the Red World. Instead, they pulled their fleet in close to protect the Cradle and Earth as they tried to stall for time to complete the Space Fold Engine. The Aggressors, a group of veteran Regulators, dispossessed Warlords who'd fallen in with CASPR, and the occasional "Reformed" Space Pirate believed that CASPR lacked the leadership to function as a wartime government. To establish solidarity with the Martian front, they dubbed themselves ARES, and drew on their contacts among other war fighters to make a bold attempt at claiming CASPR's military assets for their own.
(A victor on this conflict is left purposefully obscured, elements of both CASPR and ARES's ideologies are present in different elements of the Fleet in Exile's government. That said, that could also be because this schism never had time to develop in Exiles Among the Stars's timeline.)
The Lunar Cradle and Vitruvian Institute are somewhat outliers. Technically, the Lunar Cradle is still loyal to CASPR, though they are more interested in their own scientific pursuits. Similarly, the Vitruvian Institute is taking advantage of ARES's conflict to stress test their creations. Each is more interested in their own pursuits than politics, and would likely fold in with whoever wins with very little fuss (they, essentially, represent the XSF Battle Group, a semi-independent research branch of the Fleet tasked with giving mankind the tools necessary to fight back against the aliens).
The fate of the Martians was kind of glossed over in Fleet in Exile (though I'd laid some breadcrumbs), but they're very much still here and fighting in Worlds on Fire. The Wraiths of Mars is something I never went over in EAtS, though, again, there are connections to things that were never explored (it's also a reference to the Ghosts of Mars campaign I never ran).
[Whether or not the PC Crew is tied to either faction, or operating as independents is one of those things we'll need to sort out in The Crew thread.]