I think there are valid points on both ends of the cowl or not to cowl debate. It's true that Catwoman's look has changed over 70 years and a mask or cowl have not always been present . However, by the same token ,I think it's fair to say that to even most non fanboys Catwoman has some feature which implies she has a cat like nature such as the mask, (that Hathaway, Newmar and Kitt have), Cat Ears(Like Newmar and Co) or a cowl(Like in BR). Like Superman and Batman, Catwoman is a pop culture icon so it's hard to argue that people don't have a set image of her in their minds whether that's fair or not. It no doubt differs from person to person but it's not like she is a character like Peggy Carter who was introduced to the public at large in CA.
For me, the googles alone don't scream Catwoman because visually it could actually be Talia, Emma Peel, or Black Widow. However, with Hathaway's mask on I can buy it and say ,"Yeah, that's Catwoman". It's a subtle change to be sure , but it does make a difference imo . I think the addition of a mask separates her from just being some woman in a leather jump suit. On the bright side of this whole debate , it seems that fans don't seem to have a problem with Hathaway in the role anymore.