kohler is a little noisier, and generally will spend a little more time being maintained/repaired. Kawasaki builds a good engine as well. They are a little smoother running, slightly quieter-have a different "tone". I prefer kawasaki, but own one with a kohler. Either one, you have to stay on top of maintenance and especially the air filter. Lawn mowers run in dirt/dust and that stuff will plug and air filter quickly. I look at mine every mowing. Have to.
On the FR/FS series kawi's, the air filters have a clamp on them. Take the thumb-drive clamp off and toss it in the trash. You can't get them tight enough. Replace with a GOOD (preferably american made) hose clamp that requires a socket to tighten it up. Kohler's are mostly tool-free but I like to put some grease between the filter mounting base and the filter itself to block any entry that air could get around.
Both hold about 2 quarts of oil, both take spin on filters. Kohler's filters are a little more readily available in this area (TSC, etc).
Keep the flywheel screen cleaned off, keep the oil changed more frequently than the book specifies, keep GOOD clean air filters on them and both are good engines. Depending on the engine, they both also need valve adjustments once in a while. Kawasaki's could use adjustment a little sooner than Kohler in my experience. Just a design difference, nothing wrong with 'em.
Starters last longer on Kawasaki's. They have a automatic compression release built into the cam on most models that eases the load on the starter. Kohler does not, and in the past they've had problems with broken starters. I mean, broken-as in the shaft/gear assembly just breaks off. That's been addressed and the new shafts are larger and more robust; although I've still had to replace one or two.
I find the kawaski's a little bit easier to work on overall.
If I had a choice I'd get kawasaki. But I don't, since my old Kubota is kohler-powered, I'll just stick with what I have.