Orsino and Olivia are the alike in that both of them are playing a role of emotion that has been, at least in part, scripted for them by others. Orsino is playing the role of the courtly lover, longing from afar. Olivia is playing the perfect mourner, scheduling her time of weeping. In this both of them experience emotion, but almost for its own sake, and they seem to act as if they can manipulate emotion.
They are, however, fairly different in some aspects of their emotions. The most obvious way is that Olivia is blindsided by desire, so much so that she seems knocked off balance. She openly desires Viola (in disguise). Orsino, by contrast, proposes to Viola, but never really seems consumed by desire the way Olivia is.