As an example, Christianity (most versions of it) require a belief in original sin, a belief in a redemption by holy sacrifice, and a promise of either heaven or hell, to come in the afterlife. It is important for anyone who intends to practice this path, that they be able to not just work with, but truly embrace those concepts. These are core concepts. They are inherent to the path, and most practitioners of Christianity will not recognize as Christian, someone who does not accept and follow one or more of them, despite what that person may call themself.
Christianity does not specify or require any particular associations with the quarters. Looking at the architecture of some of the older churches, there are often figures placed in the quarters; but this is not an essential feature of the path. These are not core concepts, and there are some purists who would argue that their inclusion is inappropriate because it is an interference with the Core of the path. This is an example the kind of difference of opinion that is found among the most conservative members of any path, Gardnerianism being no exception.
The sacredness of directions. If a person sees as very important the overt acknowledgment of the quarters/directions/elements, then Christianity, which has no such overt acknowledgement, may be a poor fit. If a person does not feel comfortable with that idea/practice, then Wicca, which has a strong overt recognition of them, might not work out well.
Note that in both these cases, these general concepts can be addressed, separate from the specifics of their application to each specific religion or path. This is why so many Gardnerian teachers write their own teaching rituals and materials; this is a way to express and share their own understanding of their path, without breaking their oaths.