Hillary Clinton supports a vast majority of the policies that Obama has supported, and on that front, I don't think there will be much of a difference in their presidencies. However, Obama is a much more personable and likeable person than Clinton. She is more uptight and serious, seldom showing a fun or compassionate side.
President Obama's background was in Academia many years. The president's style comes across at times like a professor lecturing his class. Also, he has had difficulty working with Congress when it comes to much of his agenda. Hillary on the other hand is modeled a little more like her husband. She is not as likely to lecture the country when she doesn't get her way. Instead, she is likely to seek ways to find common ground with Congress.
I think Clinton has something to prove if she becomes the next president of the United States of America. She would be the first female president, so she would want to show is capable and willing to run this country how it needs to be run. Because of this, I think she'd run this country differently than the current president has, and possible better.
Regardless if you like Obama or not, he has done what he set out to do while in the capacity as president. Hilary on the other hand, can't be trusted to do what she says she will do. She has lied repeatedly to the people about several different things. She can't be trusted!