Which one would be your preference? I guess technically any watch can be worn with any strap, but I think some watches fit comfortably in either category.
Off the top of my head a really thin dress watch, say a Zenith Ultra Thin, prob wouldn't look very good on anything but a leather strap.
A "Double Duty" watch would be something like an IWC Mark XVII, which looks good on either and even has that option from the manufacturer. Speedy Pro, DJ, and Sinn 556 would fit in this category as well imo.