I would say that the two overlapped to some extent but that the Young America movement was much broader in its scope of concerns.
The idea of Manifest Destiny was really just confined to the idea of expanding the United States. It argued that the United States was destined to spread itself over the whole continent. This was based on the superiority of American government, culture, and religion over those of the Indians and even the Spaniards in Mexico.
By contrast, the Young America Movement was interested in other things. It was, for example, interested in free trade and in increasing levels of democracy both at home and abroad. It was also interested in creating an American culture that was separate from that of Europe. This is a much broader scope of interest than the idea of Manifest Destiny encompassed.