Hiding a gallery
When you Hide gallery, it removes the gallery from the thumbnail navigation. A hidden gallery is therefore not be visible anymore in the gallery index (or as a sub-gallery of its parent).
NOTE that if you include a hidden gallery in a menu, it will in effect become visible to casual visitors, regardless of this setting.
The Include images in public search option lets you control whether to include a gallery's content in the public search on your website and on search engines (e.g. Google). (Search within the hidden gallery will always be possible).