Chemex uses paper filters, and the use of paper filters needs some extra effort. A paper filter will trap even the finer coffee grounds and provides you with a more precise residue-free cup of coffee. Also, this method is slow, and the paper absorbs the oils from the coffee ground. There is a possibility that some paper may alter the flavors and taste of coffee.
In contrast, Bodum uses a permanent mesh filter and is priced lower than Chemex. Bodum brews coffee faster, and its metallic filter does not induce any flavors to the coffee. But it allows some coffee grinds to pass through, leading to sediment in the cup.
It's your personal preference to choose between them as Chemex is suitable for people who like to have fewer coffee oils and want a clearer and bolder cup. While Bodum is for people who do not like any extra flavor in their coffee and prefer the fast brewing method.