This is a great question. We need to remember that this tale is an example of an allegory, in that it is a story in which characters, settings, and events stand for abstract ideas or moral qualities. Let us focus on what happens in the tale. Dr. Heidegger invites four old friends round for dinner and offers them the chance to grow young again. Firstly, he gives them an example using the rose:
"This rose," said Dr. Heidegger, with a sigh, "this same withered an crumbling flower, blossomed five and fifty years ago. It was given me by Sylvia Ward, whose portrait hangs yonder; and I means to wear it in my bosom at our wedding. Five and fity years it has been treasured between the leaves of this old volume."
The rose is clearly very important to him, symbolising his love for the dead Sylvia Ward. However, it is this same rose, that once it has bloomed again, and then withers, that the Doctor talks about at the end, saying:
"I love it as well thus, as in its dewy freshness," observed he, pressing the withered rose to his lips.
This is highly significant, because unlike his guests, Dr. Heidegger has learnt the moral lesson of the story - we must not overvalue youth at the expense of age. Dr. Heidegger is able to accept that ageing has its own benefits, and thus he is not desperate or foolish enough to rush back to his youth and repeat his mistakes like his guests. He recognises the value of the wisdom that he has gained with age and does not want to lose it.