First off, let's define the terms: we're talking about linoleum, not sheet vinyl, though the two terms are used interchangeably these days. Sheet vinyl is flooring made out of plastic: the same stuff in PVC pipe, just with more plasticizers added. The plasticizers are known as phathates, and there's a growing movement to ban them in children's toys, because they're chemically similar to human hormones. In most sheet vinyl flooring, a big, raw sheet of off-white plastic is printed with a decorative pattern, to resemble wood or tile, and then it's embossed with a pattern. Then all that is covered with another layer of plastic, clear this time, called a "wear layer." Linoleum, on the other hand, is a completely different kind of product.
Linoleum is made by combining linseed oil, cork "flour," and sawdust, then heating up the whole gloppy mess. Color is added and then it's rolled out on a jute backing. The strips are hung in a big oven and cooked more; then it's trimmed to about 2 meters wide--that's 6' 3" for us here in the US--and shipped to stores in rolls. You can also buy linoleum in pre-cut tiles, like those shown in the image above, and a newer, click-lock floating floor system.
The key difference, aside from that there's no plastic involved, save for a thin layer of factory-applied surface finish, is that the color and pattern of linoleum go all the way through, so if it gets a light scratch, it can be repaired, a real advantage over vinyl.
Properly installed, linoleum lasts about 40 years. The key is "properly": while installing sheet vinyl is a relatively simple DIY job, linoleum is much less forgiving of mistakes. For example, linoleum can crack if it's bent too far or installed in too cold of a room.