The difference is that you as the owner are in that case not actively financing an industry that’s slaughtering other animals in order to feed your pet.
That a cat while roaming outside will inevitably kill other animals is not unethical on the cats part. It’s debatable if it’s unethical on the owners part, which is why many people nowadays discuss only keeping cats as indoor pets anyway. It is however a completely separate issue to vegan cat food.
Maybe you think vegans ask for vegan pet food because they want their cat to “be vegan”? Because if so, that’s a misunderstanding. Vegan ethics are always about our own consumption decisions and behavior. Never about those of animals. (Which is why “dO yOu JuDgE LiOnS fOr KiLliNg ZeBrAs As WeLl?” is never a good argument. We don’t.) As caretakers for our pets some of their decisions naturally fall to us. You’re always deciding for them which brand of pet food your cat will get. For example I avoid nestle owned brands, wether my cat supports that decision or not. If he was an outside pet I’m sure he would at least try to murder something occasionally. That has nothing to do with my responsibility to honor the ramifications of my own ethical considerations though. My cat is too dumb for that - literally. It doesn’t release me from the responsibility.
(He gets meat btw, he has chronic digestive problems and needs special food anyway - before anyone here accuses me of murder and torture or something.)
You know what sucks as well? Taking too many painkillers against headaches actually causes headaches. Horrible ones at that. Glad to read that you’re feeling better, but that’s a real trap many people out there are stuck in.