You moisten the skin and rub the stick on which leaves an invisible antibacterial residue. Sweat itself doesn’t smell, it’s a bacterial reaction to it. The residual antibacterial on your skin kills bacteria, hence no reaction, hence no smell.
Antiperspirant works by clogging your pores. There are rumoured health risks to this but nothing conclusive, however I don’t like the idea of clogging up my skin and definitely didn’t like the effect it had on the armpits of my shirts.
The issue with most antiperspirants is that they contain aluminium salts. It's usually a different salt than the one that forms an alum block. Both are aluminium salts, but alum blocks, like Salt of the Earth sticks and such, claim the salt they contain is too large to be absorbed by skin.
Nevertheless, there have been some reports that they are likely to speed up the onset of memory loss and related brain deseases. Maybe it's just someone spreading FUD, but I find that unsettling enough that I stopped using both antiperspirants and alum blocks.
At the moment, I use Aroma Healing's deodorants (despite the name evoking snake oil thoughts, it actually works really well... better than e. g. Old Spice sticks ever worked for me) and Schmidt's natural deodorant (Bergamot and Lime, at the moment).
They both contain baking soda and shea butter and can be irritating on the skin... unless I apply it on damp/wet skin in a single layer and then they're great and usually last me the whole day. They also last a few months each.
They can also leave marks on clothing, but unlike with some antiperspirants, these marks wash out, so my shirts aren't ruined by antiperspirants/deodorants anymore.
I can post links if anyone's interested, as the former is eBay only, as far as I can tell, whereas the latter can be found on Amazon sometimes, too.