I am native AmE. It is true that in many cases both can be used. But just as I wouldn't expect to hear, "I am ending the dishes now" meaning that I am just about done washing/drying the dishes, I wouldn't expect to hear "Our nice days are finishing now" or "February is finishing now" or "Winter will be finishing in less than a month!" It is odd but true... days, weeks, months, etc., can "start" or "begin" but they don't "finish." They "end."
And even when either "finish" or "end" works, the meaning of a sentence can differ. For example, "I am finishing the book now" can mean either that I am just about done reading the book, or that I'm just about done writing the book. But "I am ending the book now" almost certainly means that I am about to bring the book that I'm writing to a close.