The font size for \displaystyle
and \textstyle
is the same. Therefore, a difference between the two formulas cannot be seen, if the material consists of digits only.
$$2\atop8$$
$$\displaystyle2\atop8$$

But if the sum with limits is used, then the position of the limits differ dependent on the style:
$$\sum_0^\infty$$
$$\textstyle \sum_0^\infty$$

This can be used as indicator for the \atop
experiments:
$$ \sum_1^\infty \atop \sum_1^\infty$$
$$\displaystyle\sum_2^\infty \atop \sum_2^\infty$$

As can be seen, \displaystyle
does not affect the whole formula. \atop
splits the current formula in two parts, the upper and lower part. \displaystyle
belongs to the upper part only. It needs to be repeated, when it should apply to the lower part also:
$$\displaystyle\sum_3^\infty \atop \displaystyle\sum_3^\infty$$

If the whole expression should be set in a different style, then \atop
can be confined in a subformula via curly braces:
$$\textstyle {\sum_4^\infty \atop \sum_4^\infty}$$

Now \atop
is set in \textstyle
and it sets its upper and lower parts in \scriptstyle
.
$$
already sets its contents in\displaystyle
, so it makes no difference adding it.