You need to be more selective than that. Any reference to the column will turn the "UPDATE flag" on, even if the value is the same. Maybe something like this:

ALTER TRIGGER [dbo].[trgCustomer_upd]

ON [dbo].[Customer]

FOR update

AS

SET NOCOUNT ON;

if @@rowcount = 0

RETURN

DECLARE @SUPPID_changed bit

DECLARE @MANID_changed bit

SELECT

@SUPPID_changed = MAX(CASE WHEN ISNULL(i.SUPPID, '-999') <> ISNULL(o.SUPPID, '-999') THEN 1 ELSE 0 END),

@MANID_changed = MAX(CASE WHEN ISNULL(i.MANID, '-999') <> ISNULL(o.MANID, '-999') THEN 1 ELSE 0 END)

FROM inserted i

INNER JOIN deleted d ON d.CustID = i.CustID

IF @SUPPID_changed = 1

OR @MANID_changed = 1

BEGIN

DECLARE @raiserror_msg nvarchar(1024)

SET @raiserror_msg = 'Cannot update ' +

CASE WHEN @SUPPID_changed = 1 THEN 'SUPPID' ELSE '' END +

CASE WHEN @MANID_changed = 1

THEN CASE WHEN @SUPPID_changed = 1 THEN ' nor ' ELSE '' END + 'MANID'

ELSE '' END + '.'

RAISERROR (@raiserror_msg,16,1)

ROLLBACK TRAN

RETURN

END

declare @date datetime

set @date = getdate()

UPDATE tblCustomer

SET dtEdited = @date

FROM inserted Ins

WHERE tblCustomer.CustID = ins.CustID

INSERT INTO AuditDB..tblCustomer

SELECT ... FROM deleted

GO --end of trigger