They are almost always the same. There are some specific areas where it might make a difference. For example, in Manhattan in New York City there is 'downtown' and 'uptown'. So you might go down Broadway towards downtown and up Broadway towards uptown.
But it's rare when 'up the street' differs from 'down the street'. In fact, if elevation change was involved in the turn you would say something like 'Go down the street and then turn and up the hill'.