There almost always seems to be a little confusion as to these two terms. Often, too, they seem to be used interchangeably -which only adds to the confusion!
So, let's see if we can't set some things straight.
The Dark Moon
Astronomically, this refers to the period when the Moon is not visible in the sky. This lasts between 1-3 days, depending upon your location. This is the period between the end of the Waning Moon up to and including the very start of the Waxing Moon, when the Moon is again illuminated by the Sun.
Astrologically and magickally, this refers to the period when the Moon is not illuminated in any way, it is the period between the end of the Waning Moon and the start of the Waxing Moon.
Learn how to effectively harness the Dark Moon's energy for your witchcraft.
The New Moon
Astronomically, this refers to the Midpoint of the Dark Moon, the time of which the Moon is exactly between phases, i.e. Waxing and Waning, as such, all times given for the New Moon on calendars, and even on the NASA website, are the Midpoint, the peak of, the Dark Moon.
Astrologically and magickally, this refers to the very beginning of the Waxing Moon, when the Moon is again illuminated by the Sun; that very first Crescent of Moon in the Sky.
I personally prefer to use the astrological and magickal definitions as I feel they are the most accurate. Also, it's interesting to note that these are fairly older definitions to these terms, being used originally as part of maritime, astronomical, and common vocabulary.