Greg, different H numbers don't make a thread better or worse per se, they make the thread size larger or smaller. So, an H3 tap will make a "looser" thread and an H1 tap will make a "tighter" thread. This is compounded by the size of the mating male thread. Most people are not going to know the exact size (pitch diameter) of the male thread, because they would need a thread pitch micrometer, which is somewhat expensive and (for an amateur) rather fussy to properly use. Pitch diameter is basicly the distance between the middle of the two theoretically perfect V's on opposite sides of the thread. Google an image of pitch diameter and it will be easier to understand than a verbal explanation.
For most of our applications tighter is better up to a point. So what does the amateur without a thread pitch micrometer do? This amateur for example has 3/8-24 taps in H3, H2, H1 and actually H0. I can tap test threads in pieces of similar steel and then try the fit with the male thread, in this case a drum.
It is important to understand that changing tap drill sizes changes the percentage of thread or how close to a full height thread you generate, but does NOT change the size (pitch diameter) of the thread. Using a small enough tap drill can often make a thread feel tight, when in fact the tight feeling is a root crest interference between the threads.
I fear that I may be making this seem over complicated, but reading an explanation or two of thread design and thread fits will pay you benefits.
Clearer or cloudier?