I am researching this same issue. I had high school and college chemistry but am not really knowledgable. From various sources around the Net, here is what I have been able to find out about antiperspirant and deodorant active ingredients. This is all to the best of my understanding and interpretation, my apologies for any inaccuracies, corrections appreciated:
Body odor results from bacteria growing in nice warm water (i.e. sweat). You can control it by stopping the sweat and/or by restricting the bacteria. You might also not want the wetness itself.
Regular antiperspirant works by having an ingredient off a short FDA list, all of which are aluminum salts. When water from sweat mixes with one of these salts, their component chemicals break apart into “ions”, which are positively or negatively charged. The aluminum ions are positively charged and small, which apparently means that they can pass through skin cell membranes, where somehow they either cause the cell to puff up, or help form a physical plug, thereby squeezing off nearby sweat glands. No sweat also means no water for the bacteria, so no odor.
This effect is temporary, as capillaries serving the cell clean up the ions, making daily reapplication necessary for a continued effect. Theoretically the ions are eventually removed from the body in the urine, but research suggests that certain tissues (liver, kidney, brain, cartilage and bone marrow) selectively absorb it.
The FDA has recently required a new label on antiperspirants saying that people with impaired renal function (i.e. those who have trouble shedding environmental toxins) and children should not use these products. The industry is complaining due to lack of scientific evidence, but someone at the FDA seems to believe in “prudent avoidance”!
The crystal deodorants, on the other hand, typically contain potassium alum. This is the same alum that used to be used in pickles, and it is a powerful astringent. I believe it occurs naturally or can be manufactured. It has the chemical formula KAl(SO4)2·12(H2O), which includes potassium, aluminum, sulfur (as sulfate), and water.
Aluminum is apparently unusual in that it can be part of either a positively or a negatively charged clump of elements, and when alum reacts with water (“hydrolyzes”), it breaks up into its components, rearranges itself into different clumps, and the aluminum goes with a negatively charged (and large) clump. (Sorry about the hand-waving here, I haven't been able to find the actual chemical reaction.) Supposedly this means that the aluminum cannot pass through the skin cell wall, the apparent basis of the claim of safety.
Disturbingly, however, according to Material Data Safety sheets (MSDS's), when potassium alum hydrolyzes, another resulting chunk is a dilute solution of sulfuric acid (H2SO4). Sulfuric acid is, of course, corrosive, and is probably the reason for the “discontinue if rash forms” warning on the label. I wonder if it is this acid that provides an antibacterial action for these products.
I am still mystified as to why, if 30g of alum has been known to fatally poison adults, it is not more carefully labeled in its grocery store package (especially as it is reputed to be somewhat sweet to the taste.) You’d think people would also get ill from sulfuric acid if they ate too many pickles, although the residue after rinsing during the canning process is supposedly minute.
I am also suspicious of the unspecified “mineral salts” ingredients listed on the crystal deodorants. I speculate that there may be other salts similar in action to the approved aluminum ones, and that ions from these may similarly be absorbed through the skin cells. My own experience with the crystal deodorants has been that they do indeed control wetness, but perhaps they cannot call themselves "antiperspirants" because the active ingredient isn't on the FDA's list.
So what’s a person to do? My current guess is that if you must use something in addition to general hygiene and instead of odor-masking perfume, if you can use the crystal deodorant without noticeable skin reaction, it is probably safer than regular deodorant. Any trouble due to the sulfuric acid is likely to be external, immediate, and noticeable, rather than internal, cumulative, and undetectable. Now you just have to worry about those anonymous "mineral salts".
I did contact one of the crystal deodorant manufacturers, and the person who replied was friendly but not up on the chemistry.
Would I put my life savings on crystal deodorants? Not until I consulted with a quaified personal-care products chemist and had a full understanding of what I was selling. That said, given the new FDA labelling and interest in aluminum avoidance, the market for effective, safe alternative products could be good!