Mozilla's documentation on differences between Safari's and Firefox's implementation covers the difference explained below.

Safari: Storage access is granted only to the iframe that requested it, not to other iframes on the webpage and not to other subresources such as scripts and images.

Firefox: Storage access is granted to all matching subresources on the webpage such as iframes, scripts loads, and image loads.

We (WebKit) have received a few bugs where developers are asking for full storage access under the current webpage. Even so, our original intent for the API is to grant access to the specific embedded piece of content that needs it. Such a limited scope also guarantees that no other context such as other iframes or subresource loads suddenly have a change in their cookie access.

Mozillans, have you seen any issues arise from your full page scope? Did you have specific reasoning behind it?