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Coquelicot vs. pavot

While both pavot and coquelicot translate to poppy in English, they are not interchangeable. Pavot is the general, scientific term for the plant, while coquelicot refers to a specific, well-known wild variety.

Coquelicot

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Coquelicot specifically refers to the common wild poppy (Papaver rhoeas), which is famous for its vibrant red color and delicate petals. It is the flower you typically see growing wild in fields and meadows across Europe and is a strong cultural symbol, particularly for remembrance.
Au printemps, les champs de blé sont parsemés de coquelicots.
(In the spring, the wheat fields are dotted with corn poppies.)
Elle portait une broche en forme de coquelicot pour la cérémonie du 11 novembre.
(She wore a poppy-shaped brooch for the November 11th ceremony.)
J'ai peint une aquarelle représentant un bouquet de coquelicots.
(I painted a watercolor of a bouquet of wild poppies.)
La couleur de sa nouvelle voiture est un rouge coquelicot très vif.
(The color of her new car is a very bright poppy red.)

Pavot

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Pavot is the general botanical term for any plant in the poppy genus (Papaver). This includes the coquelicot, but it is most often used to refer to other types, such as the opium poppy (from which medicine and drugs are derived), ornamental poppies of various colors (white, pink, purple), and the poppy whose seeds are used in baking.
La culture du pavot est strictement contrôlée en raison de sa teneur en opium.
(The cultivation of the opium poppy is strictly controlled because of its opium content.)
J'ai acheté un gâteau aux graines de pavot à la boulangerie.
(I bought a poppy seed cake at the bakery.)
Ma grand-mère a de magnifiques pavots d'Orient dans son jardin.
(My grandmother has magnificent Oriental poppies in her garden.)
Le pavot de Californie, avec ses fleurs jaunes ou oranges, n'est pas un vrai pavot.
(The California poppy, with its yellow or orange flowers, is not a true poppy.)
Le coquelicot est en fait une variété de pavot sauvage.
(The corn poppy is in fact a variety of wild poppy.)

Summary

To summarize, think of pavot as the umbrella term for the entire poppy family, used for garden varieties, opium, and seeds. Use coquelicot exclusively for the common, wild, red field poppy. Therefore, every coquelicot is a type of pavot, but most flowers referred to as pavot are not coquelicots.