It should not be necessary to convert you date values to strings prior to comparing them. In fact, you should avoid it. The string format you're using won't sort properly anyway.
If you want to compare whole date values, convert to the DATE datetype:
CASE WHEN [YourDateValue] BETWEEN convert(date,getDATE()) AND convert(date, getDATE()-1) THEN [SomeValue] ELSE [SomeOtherValue] END
If it's not practically useful, then it's practically useless.