The Noctua NH D14 surely fits inside the Antec DF 85 case.
Question here between the two cases should revolve around looks more than size (both look ugly to me). Both cases have ample room and cooling. If I had to choose one between these two though, I'd have gone for the HAF-X.
Yes the cooler will fit in the case, but as to it being a better case than the HAF X is a matter of opinion, i actualy prefer the HAF X looks to the Antec DF85, as for cooling performance im not sure if theres much in it, could you tell us the argument for the Antec over the HAF please.
Looks are personal decisions. My previous comment summarized that both those cases are excellent and the final buying decision might come to which looks better.
Personally, I like to be a little low on the aggressive front when it comes to case designs. For instance, I like the Silverstone Raven RV02 which I'd also suggest as a viable alternative. Another alternative would be the Lian Li PC A70f.