What is the difference between the Tag Buddy and Face tag stock Gallery options, and where is the related info stored?
Tag Buddy lets you store information along with the photo (though as Andrew T. noted, this might be stored on the device and hence won't necessarily go with the photo when the photo is copied to another device).  It allows you to tag:


Weather
Location (though GPS should be in the EXIF and might be)
Whom you are with
Date (though this is in the EXIF)


The 'whom you are with' is just a list of people.

Face Tag, on the other hand actually tries to recognize faces in the photo so you can tag them, you can see that Facebook, for example, has both of these features, the ability to tag faces in photos versus being able to list 'With who..' for a photo.

Face Tag also allows you to click on the tag and do things like call or message the person.  I'm not sure if Tag Buddy has this ability.

I believe Face Tag is an Android development and Tag Buddy is a Samsung addition, which might explain the somewhat overlapping functionality, not to mention the slightly different interfaces (such as check mark versus slide)