You want to send a message at exactly 13:00:00, you have a digital clock that doesn't show seconds. What you do is that you watch your clock, and as soon as it goes from 12:59 to 13:00, you press the button. That's also how you set the time precisely by hand. With rounding, that would be 12:59:30, who wants to send a message at precisely 12:59:30 ?
You have a meeting at 13:00:00, you watch the clock to see if you are early or late. With flooring, if you see 13:00, you know you are late. With rounding, you are not sure.
It is common for digital clocks to have big numbers for hours and minutes and small numbers for seconds. If you are not interested in seconds, you can look quickly from afar and you have your usual, floored time. If you want to be precise, you look closer and get your seconds. Rounding the minutes would be wrong, because it wouldn't match the time with seconds. And you don't want clocks with a small display for seconds you may not even see to show a different time than those that don't.
And if you just want to know the approximate time and don't care about being precise to the second, then rounding up or down doesn't really matter.