After 20 years of renting, thousands of times, I will never use Budget if I can avoid them. As others have posted, Avis is the corporate, premium brand. Avis posts frequent flyer miles quickly. Avis corporate locations tend to have better customer service (your own experience may vary), more knowledgeable staff and better cars.
Same thing goes by the way with the National/Alamo family. I love National, I avoid Alamo like the plague.