My impression of a roadster, is that it is focused on performance and it's intended use isn't one of a sedan. A convertible obviously is meant for pleasure driving without the need for much storage space.
Roadsters are small two seat convertible sports cars. Some have removable hard tops too. A car like the Mazda Miata would be considered a roadster. www.buy-and-sell-car-secrets.com
A two-door, two-seater, open car without roll-up windows and frequently fitted with a dickey (rumble) seat. www.car-nection.com
A horse that pulls a bike or road wagon. Roadsters are judged working at a walking pace and at a slow, medium and full-speed trot. www.showmenews.com ))
an open automobile having a front seat and a rumble seat
buggy: a small lightweight carriage; drawn by a single horse www.cogsci.princeton.edu
A roadster is a two-seat, open car, traditionally without side windows (possibly with pluggable doortops), so that even with the lightweight convertible top raised the driver and passenger remain exposed to the elements. In modern times, the word is often used to describe a two-seat convertible without fixed window frames, especially a light-weight sportscar. Here, the use of the name 'roadster' is more a marketing than a technical one, invoking the feeling of an open-top machine for enjoyment