Select *
from tableA
inner join tableB on tableA.id = tableB.aid
                  and cast(a.date AS DATETIME) = CAST('2015-08-24' AS DATETIME)

select *
from tableA
inner join tableB on tableA.id = tableB.aid
                  and cast(a.date AS DATETIME2(3)) = CAST('2015-08-24' AS DATETIME2(3))

Select * from tableA inner join tableB
        on tableA.id = tableB.aid
        and ISDATE(tableA.date)
        and cast(tableA.date AS DATETIME) = CAST('2015-08-24' AS DATETIME)

Select * from tableA inner join tableB
on tableA.id = tableB.aid
and cast(a.date AS DATETIME) = CAST('20150824' AS DATETIME)

Select * from tableA inner join tableB
on tableA.id = tableB.aid
and case when substring(a.date, 1, 2) not in ('19', '20') then null else CAST(a.date AS DATETIME) end = CAST('20150824' AS DATETIME)