Here is my solution. Notice how I setup the temporary table, and then inserted the data. This is what you should do when asking questions like this so all we have to do is cut the T-SQL code from your post and paste it in SSMS and run it. Be sure you test your code in an empty database to be sure it runs clean prior to posting here.

Also, one of your expected values appears to be incorrect.

declare @Cust table(

CustomerID int,

CustomerName varchar(16),

AuditNum int,

AuditDate date,

ExpectedDateDifference int

);

insert into @Cust(

CustomerID,

CustomerName,

AuditNum,

AuditDate,

ExpectedDateDifference

)

values

(11111,'Cust1',1,'2015-10-01',0),

(11112,'Cust2',1,'2015-10-02',0),

(11113,'Cust3',1,'2015-10-01',14),

(11113,'Cust3',2,'2015-09-15',16),

(11113,'Cust3',3,'2015-08-30',0),

(11114,'Cust4',1,'2015-10-10',0),

(11115,'Cust5',1,'2015-09-30',20),

(11115,'Cust5',2,'2015-09-10',9),

(11115,'Cust5',3,'2015-09-01',17),

(11115,'Cust5',4,'2015-08-15',0),

(11116,'Cust6',1,'2015-11-01',0),

(11117,'Cust7',1,'2015-10-31',0),

(11118,'Cust8',1,'2015-09-28',20),

(11118,'Cust8',2,'2015-09-08',0);

--select * from @Cust;

select

bd1.CustomerID,

bd1.CustomerName,

bd1.AuditNum,

bd1.AuditDate,

bd1.ExpectedDateDifference,

isnull(datediff(day,bd2.AuditDate,bd1.Auditdate),0) DateDifference

from

@Cust bd1

left outer join @Cust bd2

on (bd1.CustomerID = bd2.CustomerID

and bd1.AuditNum = bd2.AuditNum -1)