Average people consider warts to be unsightly, embarrassing, and something to remove as quickly and painlessly as possible. Famous people do not always see the subject the same way. Some may simply not think warts to be a problem. Others may actually consider them to be a unique part of who they are.
Oliver Cromwell was one of the most well-known historical figures. Although fans of history and politics are divided in their opinions as to whether he was actually a hero or a ruthless dictator, he is equally if not more well-known for his request for his portrait to be painted "warts and all." Cromwell's death mask also contains his clearly-visible warts.
Oddly enough, many people who have no interest in the history of Britain are familiar with Cromwell's alleged quote. The quote itself continues to be widely-used, even by the many who have no knowledge of its source. "Warts and all" is a common phrase, meaning to take one exactly as he is, his imperfections included.
In more modern days, while celebrities may not consider their warts to be their claim-to-fame, the subject of their warts gives many something to comment on some more positive than others. Long-time popular movie star Robert DeNiro, while not in the media nearly as much as he used to be, is occasionally brought up by fans and non-fans alike, remarking about his quite visible wart.
While it is doubtful that not having a wart removed is due to a desire for publicity, Mr. DeNiro's wart gains publicity nonetheless. Movie star magazines and everyday "bloggers" all talk about Robert DeNiro's wart.
On female celebrities, warts today seem to be what moles were in the past often passed off as a sign of "beauty." Fans may or may not agree with this point of view.
Another widely-publicized wart is that which belonged to Elvis Presley. The story is that it was removed by a doctor prior to Elvis's military service. The fact that a world-famous singer's wart was preserved, and is still in existence thirty years after his death, says less about the scientific possibilities necessary for such preservation than it does about many Americans' unquenchable thirst for the odd and the outrageous.
Perhaps an attempt to understand it all can be traced back to Oliver Cromwell. Perhaps ordinary, everyday people have a need to see "the famous" as less than perfect, as having flaws just like everyone else. Everybody, from the famous to the average, has some type of "wart." It is certainly a better explanation than the other logical possibility that many people simply are intrigued by the macabre.