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.

What are the similarities and differences in Viola's and Olivia's circumstances in Shakespeare's Twelfth Night?

Shakespeare's Twelfth Night is a romantic comedy concerning the loves and adventures of two pairs of lovers, Sebastian and Olivia and Orsino and Viola. All of the protagonists are members of the upper classes and all are initially thwarted or misguided in their romantic endeavors, but eventually, after many plot twists, mistaken identities, and other entertaining episodes, both pairs of lovers end up happily marrying the right people.

Viola and Olivia are similar in that they are young, attractive women of the upper classes who appear to have recently suffered the loss of their brothers. Both are in love with people who seem unlikely to requite their love due to issues of gender. Both are concerned with the inner nature of their suitors rather than just wealth and power. 

The main difference is that Olivia is portrayed as living at home, enjoying wealth and position under her own identity, while Viola has been shipwrecked and is disguised as a boy and employed as a page. Olivia's brother is actually dead, but Sebastian, Viola's twin brother, lives, and is frequently mistaken for Viola. Viola is in love with Orsino, but Orsino does not know she is a woman. It is only when her true identity is revealed that Viola can move from the position of page to wife, as Orsino sums up in the following passage:

Your master quits you; and for your service done him,

So much against the mettle of your sex, ...

And since you call'd me master for so long,

Here is my hand: you shall from this time be

Your master's mistress.

Olivia's attraction to Viola as a man is resolved when she ends up marrying Sebastian, Viola's male twin, and thus the true version of what Viola was merely pretending to be. 

Compare and contrast the characters of Olivia and Viola in Twelfth Night.

Both young women approach life from entirely different directions.

Olivia seemed to enjoy her self denial.  She is self indulgent.  One only has to look at Feste in Act I, scene 5.  He proved that she was acting foolishly.  When he asked her why she was in mourning, she replied it was for her brother's death to which Feste responsed that his soul must then be in hell.  Olivia answered that it was in heaven.  Feste replied that it was foolish to mourn someone whose soul is in heaven.

When she "fell in love" with the disguised Viola, it was more the novelty of new face.  Her marriage to Sebastian was based on a misconception.  She really didn't know him.

Viola, on the other hand, embraced life and, despite the fact that she thought Sebastin had died in the shipwreck, she decided to disguise herself as a young men and dress like her brother in order to pull it off.

She also recognized that Orsino is in love with the idea of being  in love and in Act II, scene 4, she taught him about love.  Their marriage was much more realistic since Orsino was already, in a sense, in love with her as Cesario.

Both women are educated and come from noble families but each have different ideas about life and love.  

Compare the characters of Orsino and Viola in Twelfth Night.

Orsinio, the Duke of Illyria, is an intense, emotional, and passionate man who is in love with the idea of love. A statement he makes early in the play characterizes him:

If music be the food of love, play on!

Orsinio loves sensual pleasures and is often overly emotional, sentimental, and bombastic—tending to use inflated or exaggerated language.

Viola, in contrast, is self-controlled and dignified. As a person shipwrecked in Illyria, she has to survive by her wits and does so by pretending to be a man. Witty and intelligent, she has to be careful about everything she does so as to not slip up. Despite her ruse, however, Viola's manners and personality are attractive to the duke.

Orsinio, as a duke, can indulge his emotions and temper; he expects others to cater to him and overlook his excesses because of his position of power. Viola, because of her relative powerlessness, must keep her emotions under control and be attuned to the moods of those who are higher ranking than her.

What they share, however, is love. Viola is in love with Orsinio, but because she is disguised as a man, can't show it, while Orsinio, though he doesn't know it, is in love with Viola. Viola's love is much more restrained and silent than Orsinio's sentimental outbursts, but it is still love. She explains (while pretending to speak of someone else):

She never told her love,
But let concealment like a worm i' th' bud
Feed on her damask cheek. She pined in thought,
And with a green and yellow melancholy
She sat like Patience on a monument,
Smiling at grief. Was not this love indeed?