How to get last inserted id and insert it into another table using stored procedure?
Simply. Just add SET @ClientID = SCOPE_IDENTITY() after first INSERT statement. So you don't need @ClientID input parameter.   

ALTER PROCEDURE dbo.My_SP
-- Add the parameters for the stored procedure here
--@ClientID         BIGINT,
@CreatedBy        INT           = NULL,
@CreatedDateTime  DATETIME      = NULL,
@ModifiedBy       INT           = NULL,
@ModifiedDateTime DATETIME      = NULL,
@Name             NVARCHAR(255)

AS
     SET NOCOUNT ON;

     DECLARE @ClientID as BIGINT

     INSERT INTO dbo.Clients
     (CreatedBy,
      CreatedDateTime,
      ModifiedBy,
      ModifiedDateTime,
      Name,
      Active
     )
     VALUES
     (@CreatedBy,
      @CreatedDateTime,
      @ModifiedBy,
      @ModifiedDateTime,
      @Name,
      1
     );

     SET @ClientID = SCOPE_IDENTITY();

     SELECT @Name AS ClientName;

     INSERT INTO BusinessRules
     (ClientId,
      ProjectId,
      TemplateName,
      Revised,
      Serial,
      Topic,
      QualityItem,
      CriticalToQC,
      Description,
      OwnerNotes,
      RelevantDocs,
      WhereToCheck
     )
            SELECT @ClientID,
                   NULL,
                   b.TemplateName,
                   b.Revised,
                   b.Serial,
                   b.Topic,
                   b.QualityItem,
                   b.CriticalToQC,
                   b.Description,
                   b.OwnerNotes,
                   b.RelevantDocs,
                   b.WhereToCheck
            FROM BusinessRules AS b
            WHERE ProjectID = 394;