Both Viola and Olivia are women who belong to the aristocracies of their respective communities. They are "gently bred" and educated in womanly arts rather than in such masculine skills as swordplay and classical languages. Both, at the time of the play, are young, single, and attractive. Both have or had beloved brothers, but while Olivia's brother is actually dead, Viola's has merely been separated from her by a storm, although it takes them some time (and many comic plot twists) to be reunited.
That being said, the two young women differ greatly in character. Olivia has become distraught over the death of her brother, while in adversity Viola displays great strength of character and intelligence. Olivia seems more conventional than Viola and also more emotionally melodramatic, while Viola has greater self-control and sense of purpose. Despite this, Olivia at times displays flashes of insight, as when she dismisses praise of her appearance with an inventory of her beauties:
Item, two lips, indifferent red; item, two grey eyes, with lids to them; item, one neck, one chin, and so forth.
This suggests that, as a wealthy young woman, Olivia has learned to take a somewhat cynical view of her suitors, seeing that they are interested only her her appearance or wealth, and her being attracted to Viola's kindness and intelligence actually does suggest good judgment.
Viola and Olivia are both gentlewomen raised to self-esteem, manners, breeding, wealth and luxury (Olivia more so) who both recognize and accept the role of Fate in their lives. This is where their character descriptions part company. Olivia's overriding trait is emotional foolishness. Her brother has recently died after a good religious life because Feste declares him to be in "heaven." Olivia's response of mourning is to mourn for seven years with her face under a veil while refusing all declarations of love and weeping daily.
Viola, in contrast, has lost her brother and takes the encouragement offered by the Captain to hope for the best results for her brother. She accordingly develops a plan for coping with life without her brother and pursues it, even though it is certainly a plan for life that she never would have thought of prior to her loss. Both women fall in love at first sight, but Viola falls only once whereas Olivia falls once and switches once when she marries Sebastian. Speaking of which, Viola seems to have made the worse choice in her object of affection while Olivia seems to have made the better choice: Orsino is emotionally volatile and changeable whereas Sebastian is Viola's twin, which suggests her twin in character traits also.