Which is better and why, the AF Reserves or the Army NG?

OK, I assume that you are looking at the Air Force Reserve or the Air National Guard, but the questions transfer whether you are Army or Air Force.
Rank and Advancement. I see little difference between the Reserve and the National Guard at enlisted ranks at E-6 and below and Officers at Major and below. Above that it can get a bit slow in the Guard since the only positions available are within the state while you can look for another unit with a slot outside of your state easier in the Reserves. Be aware that rank and advancement can be tied to you job. If you are in a small specialty with few command slots, don't expect the rank to come quick.
There is no pay difference between the two service. Reservist all get paid the same, whether Air Force, Army, Marines or Navy.
The only difference in educational benefits is that most States offer some kind of tuition assistance for Guard members, usually free tuition to in state schools. Other than that, the benefits are the same.
Drill. I've never served anywhere that had any kind of facilities for family, so I would count on it being a weekend without them. If they haven't explained Drill Weekends, they consist of Multiple Unit Training Assemblies, a UTA is technically a 1/2 day training event. The typical drill week are either MUTA 4's or 5's, a MUTA 4 being a Saturday and Sunday, a 5 will include Friday evening. You get a full days pay for each UTA, so you get 4 days of pay for a MUTA 4.
I never found it difficult to transfer between states going to another Guard unit. It's certainly less paperwork to do so in the Reserves, but it never was a problem when my job moved me.
My best advice is to look for the unit that fills your career goals the best, whether it's Air Force Reserve of Air National Guard.
Be aware that you will have to make a plan for your children if you are deployed. Being a mother does not exempt you from being mobilized.

1. Pay is the same regardless, only the length of the drills may be different. 2. We are not a baby sitting service- unless it is some kind of family day, your kids would be in the way and a hinderance. 3. States handle medical licensing not the military- you would have to see what carries over.

What a minefield of a question. Well, the airforce has national guard units too. Guardsmen typically get paid slightly better than reservists for their drill weekends. When you're at AT you'll get paid active duty pay while you're there for both. When you're in basic and AIT for both guard and reserves you will also get active duty pay. Your real question should be, "is the army or airforce better? ". The answer is, it depends on what you want to do with your military career. Do you like austere environments and physicality? The army is for you. Do you want to do something more technical? Then the airforce is for you. Obviously there are cush army jobs and physically demanding jobs in the airforce, but in general the army is the brawn and the airforce is the brains.