For all intents and purposes, they do the same thing. The boolean version is a shortcut for requesting write access while the RegistryKeyPermissionCheck
overload allows you to specify specific permissions. But, at least right now, the only permissions are read and write but if they were to add additional options in the future the overload would allow more fine grain control. Under the hood they ultimately both ask for the same permissions and do the same work.
If you want clean code and then the boolean version is probably the easier to read.