The exact answer will vary depending on the country you're renting in (Avis exists in over 160 countries globally, so as you can imagine things do vary from country to country!), but in a general sense...
As you've stated Avis and Budget are both owned by the same parent company, and in many countries they are actually run almost as a single entity, in that most Avis locations are also Budget locations, and for the most part they even share a common fleet of vehicles.
However there are a number of differences between the two. In general, Avis is treated as more of the "business" brand, whilst Budget is more of a "leisure" brand. Many companies have corporate deals with rental car company including things like negotiated rate/conditions/etc, and in almost all cases these will be Avis, so if you're renting through your employer for a business trip, it's more likely you'll end up with Avis.
As a result of their more business-focus, Avis also generally have better benefits for regular travelers - a more complete frequent renters program (Avis Preferred, which includes better benefits the more you rent), a better mobile app, and a smoother airport location pickup process where you can go straight to your car and drive out without having to line up first.
Budget do still have some of these features, but they are generally aimed more for the infrequent traveler. Their member program (Budget FastBreak) doesn't have tiers to benefit more frequent renters like Avis does. Their pick-up process still generally requires you to interact with a person, even if only briefly, and their mobile app is less full featured than Avis.
From a price perspective, Budget will generally be cheaper if you don't have a pre-negotiated (corporate) rate - although that can vary depending on the circumstances and things like availability of coupons so it's generally worth checking pricing on both. Avis rates generally include unlimited miles (with some exceptions, such as some corporate one-way rates), whilst Budget can in some cases only include a limited number of miles per day - however this depends a lot on the specific vehicle/rate/country.
As mentioned above, Avis and Budget in many countries share a common fleet - so booking the same class of car from either vendor will potentially result in you driving exactly the same vehicle! However even then there may be differences. Avis generally offer a wider range of vehicles than budget, so if you're looking for a luxury vehicle you might find it only offered by Avis and not Budget.
Based on what I've been told my by local Avis/Budget location manager, they also attempt to prioritize better vehicles and upgrades more for Avis customers than Budget - although I don't know how strictly this is actually implemented. eg, if they have 2 rentals for a mid-size vehicle (one from Avis, one from Budget), but they only have one midsize and one full-size, then the Avis rental is supposed to get the free upgrade to the full-size car.
Avis also has more locations worldwide than Budget, although to a large extend this is because they do not operate the Budget brand in all countries. In most countries where Budget exists, the number of locations will generally be similar between the two.
From the perspective of someone looking to rent a car, I would suggest treating them both as separate companies, comparing the rates from both, and picking which is cheaper. I rent frequently from Avis and occasionally from Budget, simply because my employers corporate rates make Avis cheaper most of the time (even for leisure rentals), but if the price is better I will generally jump to Budget without a second thought as the experience is extremely similar.
