From a technical perspective, polyurethane IS a varnish. The definition of a varnish is a polymerizable substance mixed with an evaporative carrier (typically mineral spirits, and more rarely, turpentine), where the polymerization occurs by exposure to oxygen in the atmosphere. All varnishes leave a film coating behind after curing (a pure oil finish, either linseed or (pure) tung oil, does not leave a film).
The historic varnishes were usually a mixture of fossilized or dried tree resins (e.g., amber and/or copals), gum resins (such as mastic or pine rosin), linseed oil, and turpentine as a thinner and carrier. The linseed oil had a dual purpose - dissolving the resins when heated, and softening the ultimate cured finish somewhat to make it more flexible.
A modern varnish is similar - it contains a petroleum-derived resin (either alkyd or urethane) and a carrier. In the case of alkyd varnishes, some amount of plant oils are added (such as linseed, but sometimes safflower) to make the ultimate cured finish more flexible. In the case of polyurethane, the flexibility of the ultimate cured finish is controlled by the particular molecular formula of the urethane, with more polymerization resulting in a harder film, and less resulting in a more flexible (or softer) film.
Jeffrey Greene's book 18th Century American Furniture contains a good bit of historic information, and both Bob Flexner's book Understanding Wood Finishes and his recent article in Popular woodworking discuss the differences between varnishes, oils, and evaporative finishes.