Procedure with Insert Data From One Table to Another Table
I think you are just missing the @PersonID in the INSERT statement and there is no need to join to PersonLog. In other words, you should do this:

INSERT INTO
dbo.PersonLog(PersonName, Address, Mobile, FK_PersonId)
SELECT
Person.PersonId, Person.PersonName, Person.Address, Person.Mobile
FROM
dbo.Person
WHERE dbo.Person.PersonID = @PersionID  /*  Add this line */


Also, you will most likely want to surround the INSERT and UPDATE with a BEGIN TRANSACTION and an COMMIT TRANSACTION