Compare the relationships between Katherine and Bianca with Beatrice and Hero in Much Ado About Nothing.
First, there is the difference in actual blood relationship. Beatrice is Hero's cousin and, as such, is living in her uncle Leonato's house as a sort of charity. The play is not clear, but if her parents are dead, she would have had nowhere to go and live, being unable to own her own property as a single woman. Hero's father would have then taken his niece in to live with him.
Katharine and Bianca are sisters and, as such, are their father Baptista's responsibility to house, feed and maintain until the day that they each will be married. There is stress and tension regarding this in the play, since Bianca has plenty of suitors and could marry and leave her father's house, were it not for the fact that Katharine is the elder and should be married first. Katharine and potentially Baptista and Bianca are all under stress regarding this circumstance.
Katharine can be compared to Beatrice in that they both can give any man, in conversation, a run for his money. There is a sort of shrewishness in each of them that prickles at society's requirement that they be more traditionally feminine -- sweet and demure. Bianca, on the other hand, seems on the surface to be a model young lady, just as Hero does. However, by the end of Shrew, it is clear that, once married, Bianca has changed her subservient ways and appears to have taken on a bit of shrewishness. Hero, however, maintains her traditionally female demeanor throughout Much Ado.
Perhaps the largest difference in the two relationships is how protective Beatrice is of Hero, and how she appears to be willing to give anything to be "a man" and fight for her cousin's honor when Hero is wronged by Claudio. Katharine seems to have no such selfless concern for her sister. Her concern and interest is very self-involved.
These two pairs of women share some similarities, but ultimately, provide very different pictures of female relationships.