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Llover vs. lloviznar

The Spanish verbs llover and lloviznar both describe precipitation, but they are not interchangeable. The essential difference between them is the intensity of the rain.

Llover

A1
Llover is the general, all-purpose verb for to rain. It can describe any intensity of rain, from a standard shower to a heavy, torrential downpour. If you are unsure which word to use, llover is almost always a safe choice.
Está lloviendo mucho, no podremos salir a jugar.
(It is raining a lot, we won't be able to go out to play.)
Anoche llovió a cántaros y se fue la luz.
(Last night it poured with rain and the power went out.)
En la selva amazónica llueve casi todos los días.
(In the Amazon rainforest it rains almost every day.)
Si llueve mañana, cancelaremos el picnic.
(If it rains tomorrow, we will cancel the picnic.)
Después de la presentación, llovieron los aplausos.
(After the presentation, applause rained down.)
Llovía sin parar cuando viajábamos por el norte.
(It was raining non-stop when we were traveling through the north.)

Lloviznar

B1
Lloviznar specifically means to drizzle or to sprinkle. It refers to a very light, fine rain composed of tiny droplets. This type of rain is often just a mist that makes things damp but might not be heavy enough to warrant using an umbrella.
No necesitas paraguas, solo está lloviznando un poco.
(You don't need an umbrella, it is only drizzling a little.)
El parabrisas está mojado porque empezó a lloviznar.
(The windshield is wet because it started to drizzle.)
El pronóstico dice que lloviznará durante la mañana.
(The forecast says it will drizzle during the morning.)
Me gusta caminar por el campo cuando llovizna suavemente.
(I like to walk in the countryside when it drizzles gently.)
Lloviznó todo el día, así que el ambiente estaba muy húmedo.
(It drizzled all day, so the atmosphere was very humid.)
Apenas lloviznaba, pero el suelo ya estaba resbaladizo.
(It was barely drizzling, but the ground was already slippery.)

Summary

In essence, the difference is all about intensity. Llover is the general term for to rain and covers all strengths. Lloviznar is a specific term used only for a very light rain, known as a drizzle or sprinkle in English.