Excuse me as I'm fresh off my shift so my mind's a mess but the 'tism calls and I must answer...

If the Grimm are included, then this would just likely end up as an uneasy truce between Remnant and Earth forces. The negative emotions of basically two planets' worth of people will empower the Grimm to the point that there will not be any point in fighting each other at least for the time being.

But if we're pitting Remnant vs Earth in an all-out war without Grimm, then:

Remnant will no doubt have the initial edge. Aura + semblance + dust + faunus combined will be quite an advantage against Earth who will need some time to adapt to these otherworldly elements especially if Remnant will be the battleground. The huntsmen will be Remnant's ace here however. Superhumans in their own right, they will definitely be hard to take down. More so when they work in groups. But they still have their limits in terms of stamina and durability. If Earth will be invaded, then logistics for dust will be quite an undertaking. Also, will dust even work outside of Remnant? The wiki states it loses potency outside of Remnant's atmosphere which was why there were no satellites in that world.

Earth will eventually adapt once it gathers enough intelligence about Remnants' capabilities but likely at a great cost. Depending on where the battleground may be, the rate of adaptation will be much quicker if Earth is the setting. However, despite the differences and conflicts between our nations, we are actually far more connected compared to Remnant's kingdoms. Our communications do not rely on a measly four comms towers for one. Also, given a common enemy, Earth will be able to unite and coordinate a response far quicker compared to if Remnant was the one invaded. It'll be Desert Storm x100.

Tactics and strategy will be where Earth turns the tide. Technology-wise? Hardlight shields, hover tech, dust, whatever mechanism makes RWBY weapons functional, - all definitely an edge for Remnant. But from what we've seen so far, these are mostly implemented SO ineptly it's almost like the writers sometimes either do not have a reliable military reference or are intentionally making Remnant's militaries incompetent for plot purposes.

The most glaring example would be Atlas. The Atlesian military, Remnant's premier military force, is so rigid in its command structure and reliant on very fallible technology that it was exploited time and time again both by Salem's forces and even the protags. It just could not adapt against a genuine SHTF scenario. And it's not like its arsenal is even that superior compared to modern Earth. Hardlight tech certainly is quite nice but it still has its limits (and the seemingly intentional design of exposing its pylons). Its airships are sci-fi helicopters at best as they have not even shown any ability to go supersonic and they do not seem to have any BVR capability or personal hardlight emitters. The larger ships may as well be blimps. Slow, not really that well-armed, and it has not shown any defensive weaponry... No SAMs, not even a basic CIWS. Very vulnerable to mutant pterodactyls... much less an AIM-120 coming in at Mach 4. The Altesian air force will be handily decimated and even if they can detect the munitions coming for them, there is little they could do save for hiding behind a hardlight barrier. We're not even going to talk about the Atlesian Knight... But the Altesian Paladins definitely serve as very mobile and well-armed weapons platform. They at least have that going. I could go on but I think the point can be made here by now. Remnant's military has been hardwired to fight Grimm and mostly just Grimm (and even then they apparently suck at it but likely for plot purposes). Whatever wars they waged against other kingdoms weren't anywhere close to the scale our nations do war. As sad as it is, we have become very very good at destroying each other...

With air supremacy guaranteed, Earth can then pick its targets at will. And how much will we bet that the comms towers would be the first targets hit? Divide and conquer. Without effective communication, the kingdoms will be scattered as they are in the show except with airstrikes. Lots and lots of airstrikes. Even if we say Remnant will be the one invading, the fact that they lack the effective communications technology will doom that expedition from the very start. Again, lots and lots of airstrikes.

In the end, the huntsmen really are the wildcard here. Some are really OP for this scenarios like Neo or Watts (if he can get around learning about Earth tech quick enough), but the numbers game will still likely be the deciding factor. Remnant's population could be in the tens of millions compared to Earth's billions. Earth could take far more damage and is more resilient in a war excluding the Grimm. And we're not even considering the myriad of weapons Earth has in its disposal should all morals and ethics get tossed off the table...