Technically, Budget is the, er, budget brand, and Avis is the premium brand.
If both brands are operated by corporate, then theoretically, Avis should offer the better experience. I once got my hands on the Avis Budget location standards manual, and Avis has stricter requirements on their locations than Budget does. For example, IIRC, Avis managers had to open another counter terminal if the customer wait time exceeded 4 minutes, whereas Budget had no such requirement. Avis shuttles were required to be at least as frequent as the best competition, whereas Budget's shuttle frequency requirements were much lower.
Now, of course, that's in an ideal world, and who knows if every location follows those standards exactly...
I don't know if this is as much an issue in Hawai'i, since the islands are (relatively
) small and rental locations are limited anyway, but on the mainland, it should be noted that virtually all Avis locations across the country are either corporate or are licensees that use the corporate fleet and corporate computer systems, whereas many Budget branches are independent franchisees. This means that if you encounter problems with your vehicle, you should theoretically be able to stop at any Avis in the country and switch to a replacement vehicle, whereas if you attempt the same at a Budget independent franchise, you might as well be taking your Budget car to a Thrifty or Enterprise or Alamo office--they wouldn't be able to do anything for you. And if you get stuck truly in the middle of nowhere, I would have a feeling that Avis, as the premium brand that "tries harder," would be more likely to go a little bit further in getting the problem resolved as quickly as possible and making sure you're happy with as little downtime as possible, whereas Budget may take the cheapest course of action that leaves you stranded for a whole day without a car while tow trucks shuffle back and forth. Nothing to support that feeling other than my years of familiarity with the industry...
As for rewards: frankly, I think they both suck. I wouldn't choose Budget or Avis based on their rewards programs. Budget doesn't even have one, and Avis only gives you worthless weekend rewards if you're Avis First. (With the high surcharges, it's not worth earning airline miles, either, unless you find a promo that gives you a crap-ton of miles; you'll usually find those promos more with Avis than with Budget.
All of the above, though, should be trumped by any specific location reports. Honestly, there's such little difference among rental agencies these days (rental cars truly are a commodity) that regardless of the corporate standards, I'd choose a well-reputed Budget location over a theoretically-superior-brand Avis that's known for mediocrity. In places where both brands are corporate, though (which is most larger airports), all else (including primarily price, as you mention above!) being equal, I'd gravitate towards Avis, just because they at least have an image to maintain and so are theoretically more likely to kowtow to you if something goes awry.