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Manjericão vs. alfavaca vs. basílico

These three terms all belong to the genus Ocimum, but they distinguish themselves through regional usage, botanical variety, and formality. While manjericão is the broad, everyday term for common basil, alfavaca usually refers to specific wild or distinct varieties often used for medicinal purposes, and basílico is a formal or imported term often found in gastronomy and product labeling.

Manjericão

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This is the most common and standard word used in Portuguese (especially in Brazil) to describe sweet basil (Ocimum basilicum). When you buy this herb at a supermarket or order a pizza, manjericão is the word you will see and hear.
Eu adoro o cheiro de manjericão fresco na minha pizza.
(I love the smell of fresh basil on my pizza.)
Minha avó tem um vaso de manjericão na janela da cozinha.
(My grandmother has a pot of basil on the kitchen window.)
Para fazer o molho pesto, você vai precisar de bastante manjericão.
(To make pesto sauce, you will need a lot of basil.)

Alfavaca

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This term is frequently used to describe varieties such as Ocimum gratissimum (clove basil) or wild basil. Alfavaca usually implies a plant with larger leaves, a stronger and more pungent aroma, and is widely used in traditional medicine or regional stews rather than standard Italian cuisine.
Minha mãe fez um chá de alfavaca para curar meu resfriado.
(My mother made a wild basil tea to cure my cold.)
O tempero principal da peixada foi a alfavaca colhida no jardim.
(The main seasoning of the fish stew was the wild basil harvested in the garden.)
As folhas da alfavaca são geralmente maiores que as do manjericão comum.
(The leaves of the wild basil are generally larger than those of common basil.)

Basílico

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This is a more technical or erudite term, derived directly from the Latin basilicum or Italian basilico. You will mostly find basílico written on the labels of dry spices, imported sauces, essential oils, or in the menus of high-end restaurants attempting to sound more sophisticated.
O rótulo do azeite dizia que estava aromatizado com extrato natural de basílico.
(The olive oil label said it was flavored with natural basil extract.)
Este restaurante serve um macarrão com molho de tomate e basílico.
(This restaurant serves pasta with tomato and basil sauce.)
Na botânica, o nome científico da planta é Ocimum basilicum, daí o termo basílico.
(In botany, the scientific name of the plant is Ocimum basilicum, hence the term basil.)

Summary

In short, use manjericão for general cooking and gardening contexts. Use alfavaca when referring to the hardier, wilder plant often used for teas and rustic dishes. Use basílico only in formal culinary descriptions or when reading labels on refined commercial products.