Probably the most common character is the presence of the axil at the base of the tubercle, without any join to the areoles. In Mammillarias closest relatives, Coryphantha and Escobaria, the axils are separate from the areole, but have some kind of groove between them. The groove can completely joining them in some species of Coryphantha, and in others it is only partial, or only exists in old plants.
There are many other characters, including flower position, structure, and so on, but as the flowering period is short, the axil separation is the simplest first cut guide.
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Chris43, moderator