No, it is a complete Venn diagram. There are people who can only get an annulment, there are people who can only use one of the privileges, and there are people who quality for both an annulment and a privilege. It is in that last category that a decision as to which route to take must be made.
Too make matters really complicated, there is also dissolution of marriage due to non-consummation. Canon law makes this the preferred route whenever it applies:
Can. 1681 Whenever, during the instruction of a case, a very probable doubt emerges that consummation of the marriage did not occur, after suspending the case of nullity with the consent of the parties, the tribunal can complete the instruction for a dispensation super rato and then transmit the acts to the Apostolic See together with a petition for a dispensation from either one or both of the spouses and the votum of the tribunal and the bishop.