Shemini Atzeret and Rosh Hashanah are parallel in many of the kavanot and higher
yichudim.

On Rosh Hashanah, though, these are in a mode of elevation and on Shemini Atzeret in a mode of elicitation from on High downward.

On Rosh Hashanah man’s avoda is with supplication, submission and intense bitterness;

On Shemini Atzeret it is with joy.