Both contain the same active ingredient
Azadirachtin, but the
Azamax is in a more concentrated form. If you do indeed have mites you will need to spray them more than once a week, preferably every 3 days.
It is indeed a good idea to alternate miticides as mites will develop a resistance to any chemical. As a general rule of thumb you should never apply the same miticide more than once or twice in a row.
Pyrethrin based sprays can stimulate the mites reproductive cycle, so avoid them There are many
pyrethrin based sprays and foggers like Dr. Doom that are incorrectly marketed to be used on mites even though they are not effectice (they work fine for other pests though). Same goes with organophosphate and carbamate based sprays - they will only make the infestation worse in the long run and should not be used.
Insecticidal soaps and narrow range oils like Neem, Stylet, and Rosemary can all knock them back, as can dusting with Diamataceous Earth (DE). Mites cannot develop resistance to Stylet Oil and DE since they kill physically, not chemically. Unfortunately mites are very difficult to get rid of with organic methods once a population is established.