Is there any difference between "brutto tempo", "cattivo tempo" and "maltempo"? Example:
L'aereo ha avuto un ritardo per il cattivo tempo/brutto tempo/maltempo.
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In standard Italian, brutto tempo is less formal than cattivo tempo and maltempo.
The Sabatini-Coletti dictionary sets the origin of maltempo in the 15th century, but the word has become more and more formal during the time.
In a written text you would quite likely find
L'aereo ha avuto un ritardo per il cattivo tempo/maltempo.
rather than brutto tempo. In spoken (informal) language, brutto tempo would be used in most cases.
As Marcel Ferrari remarks in a comment, it can be different in other varieties of Italian, for example Swiss Italian, where it's rather frequent the inversion of formal registers with respect to standard Italian; another example is medicamento, which in standard Italian is formal for medicina (in the sense of medicinale).