When should I use Cross Apply over Inner Join?

Question

What is the main purpose of using CROSS APPLY?

I have read (vaguely, through posts on the Internet) that cross apply can be more efficient when selecting over large data sets if you are partitioning. (Paging comes to mind)

I also know that CROSS APPLY doesn't require a UDF as the right-table.

In most INNER JOIN queries (one-to-many relationships), I could rewrite them to use CROSS APPLY, but they always give me equivalent execution plans.

Can anyone give me a good example of when CROSS APPLY makes a difference in those cases where INNER JOIN will work as well?

---

Edit:

Here's a trivial example, where the execution plans are exactly the same. (Show me one where they differ and where cross apply is faster/more efficient)

create table Company (
    companyId int identity(1,1)
,   companyName varchar(100)
,   zipcode varchar(10) 
,   constraint PK_Company primary key (companyId)
)
GO

create table Person (
    personId int identity(1,1)
,   personName varchar(100)
,   companyId int
,   constraint FK_Person_CompanyId foreign key (companyId) references dbo.Company(companyId)
,   constraint PK_Person primary key (personId)
)
GO

insert Company
select 'ABC Company', '19808' union
select 'XYZ Company', '08534' union
select '123 Company', '10016'


insert Person
select 'Alan', 1 union
select 'Bobby', 1 union
select 'Chris', 1 union
select 'Xavier', 2 union
select 'Yoshi', 2 union
select 'Zambrano', 2 union
select 'Player 1', 3 union
select 'Player 2', 3 union
select 'Player 3', 3 


/* using CROSS APPLY */
select *
from Person p
cross apply (
    select *
    from Company c
    where p.companyid = c.companyId
) Czip

/* the equivalent query using INNER JOIN */
select *
from Person p
inner join Company c on p.companyid = c.companyId

Answer 1

Well I am not sure if this qualifies as a reason to use Cross Apply versus Inner Join, but this query was answered for me in a Forum Post using Cross Apply, so I am not sure if there is an equalivent method using Inner Join:

Create PROCEDURE [dbo].[Message_FindHighestMatches]

-- Declare the Topical Neighborhood
@TopicalNeighborhood nchar(255)

AS BEGIN

-- SET NOCOUNT ON added to prevent extra result sets from
-- interfering with SELECT statements.
SET NOCOUNT ON

Create table  #temp
(
    MessageID         int,
    Subjects          nchar(255),
    SubjectsCount    int
)

Insert into #temp Select MessageID, Subjects, SubjectsCount From Message

Select Top 20 MessageID, Subjects, SubjectsCount,
    (t.cnt * 100)/t3.inputvalues as MatchPercentage

From #temp 

cross apply (select count(*) as cnt from dbo.Split(Subjects,',') as t1
             join dbo.Split(@TopicalNeighborhood,',') as t2
             on t1.value = t2.value) as t
cross apply (select count(*) as inputValues from dbo.Split(@TopicalNeighborhood,',')) as t3

Order By MatchPercentage desc

drop table #temp

END