A famous character might have many reasons for wanting to keep their everyday and well-known lives separate. It might be the pseudonym that's famous (as in the case of the artist Banksy), with their true identity being a well-kept secret—probably so they can lead a normal life. But it could also be the other way around. Stephen King used the pseudonym of Richard Bachman, in part, to see if books not associated with his name would sell. Another example would be a famous children's author who decided to write a bodice-ripper and didn't want their young fans stumbling onto adult content. Regardless of the reasons, hiding this secret takes some skill and planning to pull off.

Hidden Identity (primary), Positive or Altruistic

BEING ATTACKED, BEING JUDGED, BEING LABELED, BEING UNSAFE, BEING WATCHED, COMPETITION, CONDITIONAL LOVE, CONFLICT, CRITICISM, CROWDS, DISCRIMINATION, FAILURE, HUMILIATION, LOSING AUTONOMY, LOSING ONE'S SOCIAL STANDING, LOSING THE RESPECT OF OTHERS, MEDIOCRITY, PUTTING ONESELF OUT THERE

Foregoing benefits they could reap if people knew who they really were
Not receiving acclaim or esteem for their successful anonymous efforts 
Life being more complicated because the character has to maintain two identities and keep them separate
Worrying that a loved one who knows the truth will sell the secret or let something slip
Being limited in their networking abilities

Being very picky about who knows the truth
Requiring Non-Disclosure Agreements or other legal measures to ensure others don't reveal the truth
Referencing the pseudonym as though that person is someone else
Dressing differently to fit each identity
Maintaining separate online identities
Being cautious about or limiting online activity
Limiting public appearances