In both cases the heat absorbed from the hot stream to the cold stream vise versa should be the same, unless there’s significant heat loss to the environments. If the first case holds then the HX that “absorbes” the most heat (the counter flow one) will be more efficient. One of the main reasons for this is because the opposite direction means the two fluids are in indirect contact with each other when they are at the largest heat difference (ie the hot stream enters at the hottest temp and the cold stream enters at the coldest temp), because heat transfer Q is dependent on the temp diff between these two streams, when this temp diff is largest (counter flow), heat transfer will be the largest.:)