for babyhawk, I mean the regular tie-on kind. the new Oh Snap with buckles is a whole other animal.
sorry that's not really an anwer, but I didn't want you to get all views and no replies!
and The Nurture Center Store and Resource Center 3399 Mt Diablo Bl Lafayette CA 888-998-BABY
I'd suggest purchasing from a vendor with a return policy (and bonus points if they carry both brands!) That way, you can order one... try it (being careful not to get it dirty) and if you don't like it, you can return/exchange for the other brand.
The Kozy is a taller and wider MT than the BabyHawk, and the straps are at a wider angle. I tend to find that if you have very narrow shoulders, the BabyHawk will work better.
The BabyHawk has a very firm headrest that can turn down or be up supporting baby's head. Kozy has good head support with a little extra cushioning, but it's not as stiff.
Kozy is better for high back carries with newborns/small babies because it's so tall. The ToddlerHawk works well for this but not so much the BabyHawk IME.
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Cristina, Mama to Michael , Nathan , and an angel.
Go with your gut, follow your heart.