What is the fundamental difference between the definitions of the two moons?
Answer
Their ruling definition—equinox proximity versus sequential position.
The fundamental difference rests in the ruling definition for each name: the Harvest Moon is based on proximity to the Autumnal Equinox, while the Hunter's Moon is based purely on following the Harvest Moon.

Related Questions
What primarily defines the Harvest Moon astronomically?How is the Hunter's Moon defined relative to the Harvest Moon?During which months can the Harvest Moon technically occur?If the Harvest Moon is typically in October, what month is the Hunter's Moon usually associated with?What was the primary historical benefit of the Harvest Moon for Northern Hemisphere farmers?Why did the bright light of the Hunter's Moon benefit hunters?What is the fundamental difference between the definitions of the two moons?What is the inclusive date range during which the Harvest Moon must fall?The unusually short interval between moonrises around the Harvest Moon is related to what astronomical feature?For which full moon phase would deep-sky photography likely present the greatest challenge due to lingering light?Why does the date of a named full moon generally shift earlier by about 11 days each year?