Es schüttet wie aus Kübeln.
Es regnet wie aus Kübeln.
Es gießt wie aus Kübeln.
I think they all mean it rains hard. Does their degrees of how hard differ? How do those synonyms actually differ in nuances and usages?
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If the addition wie aus Kübeln (or wie aus Eimern) is present, this already means very heavy rain - in this case, the difference between regnen, gießen and schütten are at most minor nuances.
It is different, when the wie aus Kübeln-part is missing. Then the verb would make a significant difference with regnen < gießen < schütten.
It does not really feel idiomatic to me to use the combination of regnen and wie aus Kübeln, though. Wie aus Kübeln would more likely be paired with gießen or schütten, which is the thing you usually associate with heavy rain.
Other ways to describe intensity or duration of rain would be wie aus Gießkannen (watering cans) or es gießt in Strömen (streams). A variant used in Berlin would be es regnet Bindfäden (strings), which however describes more long lasting and/or slower falling dense rain and not so much the fast falling rain where every single raindrop is quite large/heavy that gets you soaking wet within seconds.
Regnen is very general and just describes raining but doesn't specify the intensity.
Schütten by itself already means heavy rainfall but is only used colloquially.
Gießen is another colloquial term to describe normal rainfall in terms of intensity. It's often used if it keeps raining, i.e. "Heute ist es schon wieder am Gießen."
Other words for to rain are
Nieseln Very mild form of rain
Pissen Similar to gießen, but more informal (mild foul language)
I recently learned that words describing rain are very regional. After a recent move to Bavaria from the Cologne area (where we have a lot of rain) I discovered that every word I usually would use to describe rain is not commonly known here. That said, you can assume that each word can describe a different type of rain depending on where you are. For me "es schüttet" and "es gießt" would both refer to heavy rain, while "es regnet" might be any kind of rain without emphasising the degree of how heavy or light it rains.
The addition of "wie aus Kübeln" (where I grew up more commonly used as "wie aus Eimern") is again amplifying the amount of rain, although I would only use it with "es schüttet".
Interestingly enough the LVR states that Standard-German only has one word for "regnen" (and is relying on adjectives to differentiate: "leicht", "stark", "heftig" etc.), which probably supports the paragraph above, that it depends on your local dialect.