The term Family Tradition, sometimes abbreviated as 'FamTrad' refers to a tradition or teachings that are only taught to members of the family. Occasionally there is provision for someone to be 'adopted' into the Tradition, which in this case would mean an adoption into the family as well.
Hereditary Tradition refers to a tradition or a group of teachings that are only taught or shared to those who are Witches. Those who are 'of the blood'. I have heard that one way to prove this was to prove that an ancestor was burned at the stake, but that seems both difficult (for me, in America, I've had horrid luck tracing my family back to Europe at all) and chancy (seeing as how I don't trust the inquisition to actually understand what a Witch is in the first place) but I do tend toward the cynical ;).
The main point to understand here is that for a very long time, Witch was not just what one did, but what one was , by heredity as well as training. You can see echoes of this in the care with which many Native Americans revere their lore, and the hesitancy with which they reveal the true mysteries to few. The idea of race, blood, connection to the land and to the Gods and magic of that land shows up in a great many cultures.
In any case, the relevant part of this is that early on, if you weren't able to prove that you were 'of the blood' to the satisfaction of your teachers (which would be hereditaries - unless they were Family - in which case it wouldn't be a question) you simply weren't taught. The up side of this was that the student that would work to earn such teaching was probably worth teaching. The down side was that the student who couldn't 'prove' it, was not taught. Regardless of merit, which in a sense didn't need to be considered; the opinion was you were 'born' into that path, or not. End of story.
Enter Gardner. He could (allegedly) prove that he was Hereditary, and got himself accepted into a Hereditary group. He could also see that the numbers of people who both 1-could prove liniage and 2-wanted to participate in the Craft were declining. He wanted the Craft to flourish, and saw that in order to do that, a way needed to be made to allow those who were truly called as Witches, but could not prove their bloodline, to come home.
Apparently he and the coven who trained him explored this issue for some time. Eventually he was given permission to paraphrase the hereditary concepts in the materials he used. This is what he taught. This is what is considered Gardnerian Craft.
A note of caution here. In this story, "paraphrase" is meant to describe a process by which he assembled a body of lore to transmit the essence of what he had been taught. This is NOT the same usage of "paraphrase" that is seen in the eclectic community, where it's entirely possible that no two rituals done by the same group will actually be similar. He found stuff that worked, and generally stuck with it, because once he had a 'map' that would guide his students to the 'proper' destination, he had accomplished his goal. The application of his 'map' to our life situations evolves. The rituals don't.
The materials he gave remain the core of Gardnerian practice. That is how one tells if someone actually practices Gardnerian Craft or not. Additions (proper, appropriate additions, and frankly not that many of them) are permitted. Deletions are not. Every Gardnerian attending any other Gardnerian ritual is able to participate and understand, because the core remains. It is the foundation of our practice.
To some extent this is one type of disagreement between 'conservative' Gardnerians and other types of Gardnerians. Being on the conservative side of that argument, I think that the material Gardner gave is sufficient unto itself. Though one can add, I see little reason to. If you are that compulsive about having stuff in your rituals be written by you then I think you would be happier as an eclectic anyway.
Now, it is also true that there are paths that do not particularly care about heredity in any form; Christianity is one of them. Obviously that is a different kettle of fish.
The catch (of course there had to be a catch) is the exact definitions of "appropriate training" and "Gardnerian materials" which will vary according to who you are talking to and whether they are or are not actually Gardnerian (and according to who is making that call) - and even among those who are, or who say they are (which is another, and even smellier kettle of fish) there are some strong differences of opinion.