Neither are really that strong either. You could probably get a tool steel alloy with better properties.
It would be almost impossible to form it into armor without some really specialized hot working equipment. At low temperature, they're rather brittle and they melt at 2500-3000 C, so you'd need to get to at least 1200 C (~2300 F) before they started to soften. Or you'd need to do it all by powder metallurgy.
Dense and brittle are not good properties for armor. Dense means it will be very heavy, brittle means it is going to fracture.
In addition to the details given by the other answers, consider this. A chest plate made of iridium, say 1' x 1' (approximately 30 cm x 30 cm), which is only 1 mm thick, would weight somewhere in the neighborhood of 5 pounds. Iridium may be slightly less rare than osmium, but it is still very rare. Rare means expensive. That one chest plate would have a value of around $75,000 at current iridium prices. That isn't even factoring in how much material would be lost during the labor-intensive processes needed to work the metal. You could easily be looking at $100,000 just for a chest covering - and with no guarantees about how well it performs. A millimeter thick probably won't stop a rifle bullet. I certainly hope this guy is very wealthy.
Ballpark, I'd say a half million dollars or so to cover your torso in 1 mm of iridium, probably a couple of million to do the whole body. This suit would weigh somewhere in the neighborhood of 50-100 pounds at that thickness (which again, I have no reason to believe could stop a bullet, I just picked it for calculation).
You say "but iridium isn't as rare, while that seems to be true, it still is very rare. While both are hard (probably a useful property for armor) they both are also brittle, a bad property for armor. They are both also both very dense, while high density may have some advantages for some types attacks on fixed targets, overall is is a serious disadvantage for body armor.
I'd be inclined to say Osmium is better because there is no chance of acquiring enough for the job so you will be forced to abandon this rather impractical project before it starts. However, Wikipedia says:
"Osmium reacts with oxygen at room temperature forming volatile osmium tetroxide. Some osmium compounds are also converted to the tetroxide if oxygen is present.[48] This makes osmium tetroxide the main source of contact with the environment. Osmium tetroxide is highly volatile and penetrates skin readily, and is very toxic by inhalation, ingestion, and skin contact"
So I can't recommend Osmium, I guess you are stuck with Iridium. "PRICE IS NOT A PROBLEM" You do realise this project will cost millions (plural) right?
As the other answer stated, osmium is out due to toxic properties.
and 1 mm is far from enough, I'd say 10 mm at least. that brings the price into the tens of millions of dollars and the weight to 1000 pounds. Totally not practical.
Plus a few millions in research and staff to do the machining.
Ok so im working on a project i got interested in. It's making the strongest armor possible i got the project from a military enthusiast and a mineral collector. He wants to make either body armor or a whole suit of armor. The problem is he wants to make it out of irridium or osmium but osmium is rarer and harded to get but iridium isn't as rare and he wants osmium if at all possible so are either of these two minerals viable options for body armor. PRICE IS NOT A PROBLEM. I just need to know where to get it or how to make it and if they could work or if they are too toxic or dangerous.
