Most people don't notice New Moons. It's easy to see when the Moon is full, but the only way to tell when a New Moon is happening is during an eclipse, or by referring to an almanac or using Obliquity's Moon Phase calculator.
To Wiccans, the second New Moon is called the Black Moon, and any magic worked during that period is deemed to be especially powerful.
Of course, the chances of two New Moons falling within one calendar month are just the same as two Full Moons, but because New Moons are generally invisible, most people tend not to notice the occasions when a month has two of them.
That's not to say that New Moons aren't important to non-astronomers. To the world's Muslims, the date of New Moon is of great interest, since the Islamic calendar is governed by the phases of the Moon: the start of each month is marked by the first sighting of the new crescent Moon.