Both are question answer website. Yahoo offers no follow backlinks from their website that help to redirect traffic to your webpage. This helps to build traffic to your website.
Yahoo answers has a greater emphasis on the question, and Quora on the answer. Quoran questions must be 250 characters or less, and cannot have details, only links. Answers however can be very long, and have pictures, Italics, bold, lists, and sourcing. Also Quora has a much greater emphasis on community than Yahoo.
According to quora: Request Answers is a way to connect people with questions to people who can answer them. At the same time, Yahoo answers has no such social element. ... Quality and Quantity: Quora managed to get both quality AND quantity. Yahoo Answers failed in the quality stakes.
Yahoo answers is becoming more of a comedy site than a real question and answer forum. Quora is for serious answers, while yahoo often has funny and irrelevant ripostes, with lots of trolling.