Hiya Ita-San, I think you are correct assuming that 'insurance' and 'assurance' can sometimes have pretty much the same meaning with 'assurance' being more used in BrE and 'insurance' in AmE. However, 'to assure' has also a different meaning in that it is most of the time used with the idea of stating something with great confidence, or taking measures to make something be true : "I assure you that this problem will be solved by tomorrow" = "I guarantee you that this will be fixed tomorrow" eg "I will make sure this is solved". To make it even more confusing, there is also 'to ensure' which is also used to say 'to make sure that'...
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