Well, there are two different actions there. The Mail Red button does NOT shut down the application, just pushes it into the background. You can tell that it's still running because on the Dock you will see a little dot below the icon for Mail, showing it is still running. But if you truly quit it via the menu, the dot goes away and the app is NOT running anywhere. With reminders, the app actually closes with the red button (the dot goes away), but because reminders is a "reminder" system, there is a background task running that watches the clock to alert you to any date/time you have set for a reminder. With Mail, you don't need a "reminder" at a time or date, so it doesn't have any background task running when you quit it.
I don't think there is any reason for the two apps to behave the same in this circumstance. One has a background process to watch the clock, the other does not. One needs to watch the clock, the other does not.
So, I guess the suggestion is to not quit Mail. There really isn't any need to do that, it runs quite nicely in the background and the alert works perfectly.