In astrology, all physical affects are provided by Moon, and the first rule is to find tithi when Moon is strong. Moon is generally strongest on full moon day and its strength gets proportionately reduced as it moves toward Amavasya or New Moon day. As such Moon on 13th or 14th tithi of dark fortnight is very close to Amavasya or New Moon day and only a tiny crescent of Moon is visible. Auspiciousness, as per Vedanta Jyotish, requires that moon must be at least 60 deg. away from sun, i.e. it should not have crossed 10th tithi of dark fortnight.
The reason being, that devotees worship Lord Shiv and Mata Parvati throughout the night and prefer to physically view the moon on the next morning immediately after Mahashivratri. And if one wonders why Muslims start there month after physically viewing the new moon , the reason is the age old tradition of viewing the LAST CRESCENT of MOON on the next morning after Shivratri and Mahashivratri .