Pagan is a general term while Wicca applies specifically to those belief structures that are descended or adapted from Gardnerian Wicca, which was founded by Gerald Gardner in the mid 1900s. Gardnerian Wicca is the first formal, Wiccan tradition that was known about publicly through media interviews and the publication of books by the members, with Gardner being the first to publish. There are other and newer traditions of Wicca that claim different, or even no, lineage at all.
Think of it this way, Paganism is a tree of which Wicca is a branch; in much the same way that Christianity is a tree of which Lutheran is a branch.
But it is not always just to label a Wiccan as a Pagan...
Though it is controversial in both the Pagan and mainstream religious communities, some people believe that Christian, Jewish, Muslim beliefs can be melded with Wicca to create Christian Wicca, Jewish Wicca, less often Muslim Wicca, so it is even unfair to say that all Wiccans are pagans, some believe themselves to be very much Christian, etc and would be upset to be identified as Pagan.
While this is controversial, it's not a new controversy, through the Middle Ages, many learned men (John Dee for one) pursued magic and alchemy and worked with calling angels and demons though they considered themselves to be very much Christian. The tradition of mysticism in even mainstream religions is a very old one that continues today.
I have often explained it this way to my students and others who ask: Wicca is to Paganism as Catholicism is to Christianity... A sub-section of a larger belief system. Not necessarily the largest, or the one with the most followers, but the most recognisable to the outside world. And in my experience the ones with the most rules.