Until a few years ago, most lenses were made of crown glass, which is heavy and has poor impact resistance. When it does break, it shatters into sharp shards, which could do serious damage to the wearer!
Today, most opticians would recommend plastic lenses, because they are about half the weight of glass lenses, and very impact resistant- if they do break, they snap into large blunt(ish) chunks, which should not cause the wearer serious injury.
Ultimately the thinnest lenses available are glass- special High Refractive Index materials, allow the lenses to be made thinner. The higher the refractive index of the lens, the thinner it can be made.
Standard Crown glass has an Index of 1.523 while plastic lenses (CR39) have an index of 1.49- that means that a glass lens would be slightly thinner.
Until recently, if someone with a high prescription wanted their lenses thinned, glass would have been the only option- commonly available indices for glass would be 1.6, 1.7, 1.8 and the thinnest available is a 1.9 lens. Now plastics are catching up, with higher index lenses available- common values are 1.6 and 1.67, with 1.74 being the thinnest. Plastic lenses are now available in photochromatic materials (where the lens darkens when in bright conditions), which until recently, was again a glass only option.
Most lenses are made in plastic for good reason, but still sometimes, for the thinnest possible job, a glass lens may be used. Glass lenses can be heat or chemically treated to improve their impact resistance- called toughening, but still the most impact resistant lens is plastic, and Polycarbonate and Trivex are the best of all.
Plastic lenses are softer, and therefore more prone to scratching than glass, but scratch resistant coatings can be applied to reduce this weakness. Overall plastic is usually the better option