The difference is that a furuncle is a purulent inflammation of the hair follicle, the sebaceous gland that surrounds this follicle and the nearby connective tissue, is often caused by all Staphylococcus aureus. Here, and mastitis is an inflammation of the mammary gland itself, is also caused by Staphylococcus aureus and Streptococcus ... These are different things and it is hard to confuse or misunderstand them, so consider all this information when you go to the doctor)
Believe the textbooks, I guess. I think that not only doctors say different things, but you understand them differently. Or they translate differently from medical into Russian.
To put it simply, a furuncle is an abscess inside the skin, limited, small Mastitis is an inflammation of the gland (what's inside, under the skin), maybe purulent, non-purulent, even gangrenous ... there may be an abscess (a bag of pus under the skin or in the depths), maybe spilled pus, not collected in a bag