At the beginning of the play, both characters at least profess to abandon themselves to their emotions. Orsino embraces the extreme depth of his love, praising the "spirit of love" for how much it can devour. Valentine says that Olivia's servant Maria has told him that Olivia will walk around her room twice a day crying to mourn her brother. When we actually see Olivia, however, she at first seems much more dignified and restrained than Orsino. She doesn't reveal to "Cesario" how she feels about him at first. When she does in act 3, her oath is by all modest things: "Cesario, by the roses of the spring / By maidhood, honor, truth, and everything / I love thee so that maugre all thy pride / Nor wit nor reason can my passion hide." She is not able to contain herself, but unlike Orsino, she believes that she should. ("Maugre" means "despite.") At the end of the play, when she sees Orsino in person, she reminds him of the rules of noble behavior:

ORSINO What shall I do?

OLIVIA Even what it please my lord that shall become him.

This statement is Olivia calling on Orsino to behave in a dignified and controlled fashion, which he certainly does not, as immediately afterwards he threatens to kill Cesario. Whereas Orsino is committed to expressing the excess of his emotions, Olivia believes in a noble restraint, even if she is not always able to follow through on that belief.

Duke Orsino begins the Shakespeare's Twelfth Night pining for Lady Olivia, who will not let him woo her because she is in mourning for her brother. For her part, Olivia's grief is quickly put on a back burner in favor of romance the moment she meets Viola disguised as Cesario. So one way in which these characters can be shown to be similar in how they experience strong emotions is how quickly and deeply they fall in love. Each of them have had very limited encounters with the objects of their affections but can still monologue about that affection at length. They are both prone to pining.

They also both refuse to stop pursuing their suit (i.e., wooing their crush) even in the face of repeated rebuttal. Orsino is not allowed to visit Olivia so he's sneaky and sends Cesario in his stead. Olivia is also sneaky in professing her love, sending her ring after Cesario and pretending that it his own that he left behind as a message of intent.

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.