Because the month tells me more about how far in the future something is. If I have an appointment on the 12th of July, there's not much information in knowing it's on the 12th. 12th of what? But it's in July, so between 1 and 2 months in the future. If I need more info, then I'll pay attention to the day. So in order of information given.
Historical dates are similar, except I really just need (roughly) the year, and then a month if that's relevant. Knowing the exact date of a historical event is just showing off. But if you know the month, you know what season it was, what the weather was probably like. Was it planting/growing/harvest time? You can guess at a lot of things with just the month.
Sure, but if I tell you the month, you still know what part of the year it is. If it's sunny, or if it's rainy must mean something to you.