Mostly, no, they're not duplicates (but if you downloaded an extra copy on Photos, then yeah, they are both using storage).
The Gallery app and Photos just show what's on your phone. If you delete a file from your phone, it will disappear from the Gallery, but MIGHT still be in Photos, although the file isn't taking up space on your phone.
Photos will show you all synced images on your cloud account, whether or not they've been deleted from your phone (the 'preview' images do take up a little space on your phone as cached thumbnails, but the actual file isn't downloaded on your phone unless you specifically ask to download a copy). If a picture is deleted from your phone but kept in the cloud, Photos will still show it, but the picture isn't taking up space on your phone because it's just showing you an online copy of it. Does that make sense?