Excellent question. I would want to argue that the central conflict within this wonderful allegorical story concerns the conflict between the foolishness and beauty of youth and the wisdom and decrepitude that comes with age. Dr. Heidegger, by conducting his little experiment on his guests, wishes to discover if, when given an opportunity to live their lives over again, they are able to learn from their mistakes and not fall into the same patterns that resulted in them being so unhappy in their old age. Note the response after they have quaffed the magical water:
Youth, like the extremity of age, had effaced the strongly marked characteristics of middle life, and mutually assimilated them all. They were a group of merry youngsters, almost maddened with the exuberant frolicsomeness of their years. The most singular effect fo their gaiety was an impulse to mock the infirmity and decrepitude of which they had so lately been the victims.
Note how Hawthorne creates an opposition here between youth and old age, and how he clearly presents the guests as not having learnt anything from their mistakes. They each fall into their old vices very easily and quickly, with the men vying for the attention of the preening Widow Wycherly. The way in which the guests go in search of the Water of Youth at the end of the story show that they have not learnt anything from the lesson that Dr. Heidegger has taught them. Dr. Heidegger never would drink from the Water of Youth, as he has managed to learn from his experiences in life and appreciates the wisdom he has thereby gained. His foolish guests, on the other hand, have no such perspective. Thus the central conflict in this story is between youthful foolishness, and old age and wisdom.