The answer to your question is really quite lengthy, as it involves, among other things, equipment, mission and history. Quite a few Air Guard units existed prior to WW II as part of the Army National Guard. All Air Force Reserve units were formed after WW II.
Beginning in 1943, high-level talks were held in Washington about the future of the National Guard, and the emerging Air National Guard. It was decided then that the reserve units would provide trained personnel as fillers or reinforcements, to be used by existing active duty Air Force units. These units would train under the direct control of the active-duty forces. This is one of the reasons why when you look at the history of AFR units at the beginning of the Korean War, you see that many units were activated and then quickly inactivated-they had accomplished their mission, and their personnel were assigned to active-duty units.
The Air Guard, on the other hand, had political connections and state responsibilities that were not particularly appreciated by active duty leadership. The Air Guard’s role when called to active duty (federalized) was to provide trained units (not simply personnel) to be deployed with the active duty forces.
A lot has happened since the mid-1940s description above, and at certain times it is difficult to understand why two reserve forces exist today. One of those times for me is when I land at Minneapolis/St. Paul only to see an Air Guard C-130 unit at one end of the field, and Air Force Reserve C-130 unit at the other.
But there are differences, and they can be significant. Many personnel in the Reserve serve in a status known as Individual Ready Reserve, while most Air Guardsmen do not. Around 1972 the Air Force developed the Associate Wing concept for airlift wings operating the big heavies. The Air Guard had few heavy airlifters by that time, (though they had many earlier) and did participate in this program. Another example of existing differences is that the Air Guard has many communications and tactical control units for which the reserve has no equivalent. The Air Guard operated many Tactical Recon fighter units up until about ten years ago, while the Reserve had none, and as has already been mentioned, the Air Guard operated 100% of the Fighter Interceptor units.
As far as equipment is concerned, I am well prepared to argue that neither the Air Guard nor the Air Force Reserve was ever issued crap, as one respondent has suggested. Many Air Guard A-10 units received their aircraft directly from the factory. For that matter, when researching individual aircraft histories, you can find aircraft that served in the Active Duty Air Force, and then were swapped back and forth between the Guard and Reserve. In the case of certain RF-4C Phantoms, after they served with the Active duty Air Force, they were issued to the Air Guard only to be reintroduced to service in the active-duty Air Force much later. And not as crap.
Now I have a question. Twenty years ago there was no RAF reserve, but there seems to be one today. Can somebody step up and outline the more or less recent development of this force? I would like to know what types of aircraft they operate and do they share these resources with the active duty RAF, or do they "own" their aircraft?