Here's a few of the things people consider to be advantages of linoleum:
durability- on a whole different scale than vinyl. You could expect linoleum to look brand new for 10 yrs or more, look good for 40+. An old department store in my town has a 60 yr old floor that looks great. When I last saw my grandparent's house in 1975, the 1916 kitchen floor was still doing the job. There's no wear layer, colors and patterns go all the way through the material. Won't melt a hole from a dropped cigarette or ember; gouges, cuts, nicks are easily and almost invisibly repaired.
All "natural" ingredients- no petrochemicals, no environmental toxins involved from manufacture all the way through disposal. In fact it has a natural bactericidal quality that makes it a flooring of choice for hospitals, etc.
And of course some people think it just looks better than vinyl. Incredible colors and patterns are available. Older patters are great for restoration looks.
Linoleum wasn't made in the US from about 1974-1998 or so. Now it is more easily available again, Armstrong is the biggest source and Forbo for imported.
Modern adhesives are used with today's linoleum, and seams are often heat-welded; in the old days it was common to install it floating loose to freely expand and contract with 1/2 inch gap under the baseboards or even staying well shy of the wall as sort of an area rug effect.
Well I could go on but I'm sure this is already more than you really wanted to know.