I have wondered the same thing. I usually get crumble. The mash seems like it gets waisted easier due to being a powder. And I worry the pellets are harder for them to eat. Sometimes they are rather big pellets. But I have no facts on this.
crumbles are just pellets that have been broken up, My opinion is that the larger pieces get wasted less. A chicken doenst need to be able to chew the piece, just able to swallow it. Mash often can seperate and your chickens will not get all the nutrients at one time, but I would suspect that in the end the chickens would get enough of what it needs, chickens have been fed mash for years, and they are still around.
So advantage to crumbles or pellets is they get nutrition in every bite, and there is less waste.
Pellets here too. I've read many posts where people say they've had to go back to mash or crumbles because their chickens didn't "like" the pellets. An animal won't starve itself if appropriate feed is available to them. By appropriate I mean unspoiled and of the correct type for the animal.