To use the Remote Play feature, the same PSN account must be linked to both systems.
If you need to switch the PSN account on your PS4, select [Power] > [Switch User] on the home screen.
Only one PSN account can be linked to a PS Vita. To link a new account, you will need to restore you PS Vita system. You can restore the system software to its default condition and delete all of the data in system storage by selecting [Format] > [Restore This System], and then follow the on-screen instructions to complete the operation.