>>>Why do you need a union and the next select twice (which can be avoided with a cte I think)
>>>
>>>

>>> select TestScore from (select *, row_number() over (order by TestScore, StudentID)  as Row from @StudentsScore) X 
>>>

>>
>>UNION and BETWEEN produce the same results. I don't think there is a difference in execution plans (will verify). I agree, that single statement is easier to maintain.
>
>(1) twice the same select statement makes the query run twice - second time probably faster - but twice nonetheless
>(2) I doubt that Union and between produce the same result - try to select the same record twice with between

I changed a bit, but still BETWEEN worked better.

set statistics time on
set statistics profile on
set statistics io on
print 'Creating cte'
;with cte as (select *, row_number() 
over (order by TestScore, StudentID)  as Row from @StudentsScore)

   select avg(TestScore*1.0) from (select TestScore from cte where Row = (@TotalStudents+1)/2
    union all select TestScore from cte 
    where Row = (@TotalStudents+2)/2) Y 
print 'Finished with the UNION select'
print 'Creating cte'
;with cte as (select *, row_number() 
over (order by TestScore, StudentID)  as Row from @StudentsScore)

   select avg(TestScore*1.0) from cte where Row between (@TotalStudents+1)/2 and (@TotalStudents+2)/2
print 'Finished with the BETWEEN select'   
set statistics io off
set statistics profile off
set statistics time off