If anything, 'head off' is slightly more upbeat and enthusiastic than 'set off'. You could 'set off 'with a heavy heart or a light one, but if you 'head off' there is a hint of jollity and extra purposefulness/determination.
The basic meaning is the same in this case, but there is a difference in both tone, as manon33 has noted, and possibly in emphasis as well. To head of on an adventure means to begin an adventurous journey, with the emphasis more on the adventure than on the initial departure. In contrast, if you say set off on an adventure, there is more emphasis on the departure, the act of beginning the adventure.