However, was the peel ever mentioned as perhaps being one of the most important aspects of a apple's benefits to health? I don't remember my mother telling me to eat the skin as a child, so have I been missing out?
Apple peel contains Ursolic acid. This is the natural chemical that gives an apple its natural waxy surface shine.
Ursolic acid is a natural acid, that recent tests have shown may be capable of inhibiting various types of cancer cells by inhibiting the STAT3 activation pathway, and human fibrosarcoma cells.  
Ursolic acid is not only found in apples. It is also present in many herbal plants, such as basil, peppermint, rosemary, lavender, elderflower, orgenao, and thyme; and other fruits including bilberries, cranberries, and prunes.
However, apple peels contain large quantities of ursolic acid and other related compounds that may help bring out its properties. It has also been implicated in preventing muscle weakening, as well as having a general health responsibility in a daily healthy diet including weight control, balanced cholesterol, and balanced blood sugar.
“The importance of apple peel was discovered after Dr Adams, a U.S. expert in how hormones affect the body, set out to find a drug that stops muscles from wasting, keeping pensioners strong as they age and cutting their risk of hard-to-heal fractures.”
The exact amount needed for these beneficial properties to be potent enough to help actual muscle wasting conditions, for example, is not yet quite known. But it may be that if large doses are needed, it may be possible to take it in a concentrated form, a natural formula tablet, as a supplement in the future.
In the meantime, eating apples as part of your daily diet could help. Remember to go for organic produce, and avoid chemical or GMO intervention.
 Shishodia S, Majumdar S, Banerjee S, Aggarwal BB (2003). "Ursolic acid inhibits nuclear factor-kappaB activation induced by carcinogenic agents through suppression of IkappaBalpha kinase and p65 phosphorylation: correlation with down-regulation of cyclooxygenase 2, matrix metalloproteinase 9, and cyclin D1". Cancer Res. 63 (15): 4375–83.
 Pathak AK, Bhutani M, Nair AS, et al. (2007). "Ursolic acid inhibits STAT3 activation pathway leading to suppression of proliferation and chemosensitization of human multiple myeloma cells". Mol. Cancer Res. 5 (9): 943–55.