They get world wide death threats. William has full security and lives behind the palace walls so a little different.The police is not Harry's private security force. As Ant noted, credible threats can be shared with police, and private security can handle day-to-day stuff.
Harry can bring his family for a low-profile visit to see his father and grandmother. Unless he is so freaked out about security that he doesn't believe he and his family would be safe on private, well-secured royal properties. In which case, what would police protection add?
Some of the coverage specifically refers to photographers allegedly chasing Harry' car during his visit for the unveiling of Diana's statue as part of the reason for the request (see here). That's normal paparazzi stuff and has nothing to do with who he married.
William seems to have a much healthier approach to media coverage of his family, and they appear to be doing well with how much exposure the kids get. I don't think Harry's antagonistic approach is doing him any favors.