I am often complimented on my choice of outerwear in the Seattle rainy season, and I hate to respond to a well-meant compliment with a correction. So I usually let all those "Nice trenchcoat!" comments slide and just say "Thanks!" But as a public service, let me lay it out for you so that you don't make the same mistake. Here we see David Tennant as the Tenth Doctor wearing a classic example of a trenchcoat: (Click for a larger version.)
The trenchcoat is a long waterproof coat, traditionally made of gabardine. The term originated in the trenches of the First World War, due to the popularity of this style of coat amongst officers in the British armed forces. The trench coat is not merely a functional warm raincoat but also stylish, with long wide lapels and decorative buttons. The trenchcoat is often belted and might be tailored in at the waist, particularly for women's trenchcoats.
A duster is also a long waterproof coat that is often referred to as a "trenchcoat" -- but as you'll see, it is quite different in its details. Here's the duster I wear, an Australian-made Driza-Bone:
Note the lack of decorative elements, the flap over the closure, the no-lapel collar (which clasps shut, completely enclosing the neck if necessary) and the built-in extra rain protection on the shoulders. (*) Dusters are typically made of oilcloth and are built for handling the practicalities of herding sheep in the rain, not for style (**).
Not shown in this view: the interior includes straps that let you attach the bottom of the coat to your legs, so that it does not blow around when you are on horseback. Also, the back is cut in such a way that you can cover both your legs and the rear portion of the saddle with the coat. I usually take the bus and not a horse to work, but still it's nice to know that options are available should I need them. These practical elements are usually not present in trenchcoats.
Right, glad we got that sorted out. Next time: What is binding, and why is it always either early or late? Can't it ever be on time?
(*) Duster manufacturers always hasten to point out that the shoulders are already waterproof; the extra layer keeps your shoulders warmer by shedding rain more effectively.
(**) There are, of course, some dusters built for style; if you watch the "Matrix" series of movies you'll see the heroes wear an assortment of extremely stylish dusters and trenchcoats both.