Let's see...
Both Godzilla and the tarrasque have a ridiculous healing factor, meaning that most wounds dealt to them will heal quite quickly and neither of them will kill the other, so, as the OP states, the battle ends when one of them KOs the other and damages them to the point that any other creature would be considered dead.
The tarrasque is 50 feet tall, 70 feet long, and weighs 130 tons. The smallest version of Godzilla is 164 feet tall and weighs 20,000 tons, while the largest (and most current) version is 328 feet tall and weighs 60,000 tons.
Both are slow and lumbering (but capable of short bursts of speed), although due to Godzilla's much greater size, he does technically move faster (as each step brings him farther then the tarrasque's can). Also, at least one version was able to jump several thousand feet in the air.
While the tarrasque is just a dumb animal, Godzilla usually displays some degree of intelligence. Exactly how smart he is varies: while most versions are smart like a wolf or orca, having an ability to strategize and trick opponents in battle, the version of the cheesy 60s and 70s movies was actually sentient, having intelligent conversations with other monsters in a monster "language."
They are also both very durable, with Godzilla being unaffected by tanks and missiles... and nukes and meteors, and the tarrasque being at least immune to non-magical medieval weaponry and having a carapace that reflects all energy weapons, although it only covers his back. I'm going to assume that they are both capable of piercing each others skin.
Although I'm not sure how strong the tarrasque is, I'm almost certain it's not enough to lift 20,000 tons, let alone 60,000. Meanwhile, Godzilla has thrown a 30,000 ton monster over the horizon ( admittedly, with the help of some leverage).
And finally, perhaps the most important factor, their methods of attacking each other. The tarrasque only has mundane melee attacks (claws, teeth, horns, and tail). Godzilla has those (minus the horns) and his Atomic Ray, which causes a lot of harm even to most other giant monsters. Even if tarrasque is unaffected by the heat aspect of the attack (I'm assuming immunity to fire applies to heat in general), it's not immune to radiation or pure kinetic energy (the ray causes explosions when it hits the ground and buildings and often pushes back lighter monsters). Also, several incarnations of him have an Atomic Pulse (which is basically an unconcentrated Atomic Ray that radiates a short distance in all directions from his body) and a Hyper Spiral Ray (which is like the atomic ray x50). And if you want to really stretch it, he has also flown by using his Atomic Ray like a jet and absorbed lightning to somehow magnetize his body, but I wouldn't count those since they both come out of nowhere and are only used once in the series (also, I'm pretty sure at least one of them was from a movie that was originally intended to be for a different monster, and they didn't write it out of the script when it was changed to a Godzilla movie).
All considered, I'm gonna say Godzilla has this one in the bag. Now, don't get me wrong, I love the tarrasque, it's my favorite DnD monster, but it's basically Godzilla without any special powers and scaled down so that it can be fought by a party of adventurers, with the only added advantage being its reflective carapace, which, as I mentioned, only covers its back. They don't call Godzilla the "King of the Monsters" for nothing.
Tl;dr: Godzilla wins.
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