Denny, First, make sure that you and your installer are talking about the same material. Vinyl and linoleum are actually two different materials. True linoleum, made of all-natural materials like jute, is different than vinyl, which is plastic-based.
Most people, unfortunately (including most flooring installers) use both terms interchangeably to describe vinyl. Linoleum was the common rolled flooring of much of the last century, and is enjoying a revival of interest because of its period look, its durability and its "green" environmental qualities. See Jane Powell's recent book on the subject, titled simply, "Linoleum." The L.A. Times' Home section, Old House Journal and This Old House Magazine have all had recent articles on the resurgence of linoleum.
If you want to compare both materials side-by-side, visit Linoleum City in Hollywood. They carry many patterns, some old and some new. They will also be able to direct you to an independent installer who can do a simple installation, or do intricate inlays of different materials and patterns.
As for the subfloor, if you have doubts about changing it, I would suggest getting a second opinion from another flooring installer.