Quantcast focuses mainly on the nature of a website’s or mobile app’s audience. This includes the user’s demographics and psychographics, other sites a users tend to visits, what businesses and industries they represent and cross platform (online, mobile, and app) engagement and retention. From this information, all the websites measured by Quantcast are put in order from 1 to 100,000 (typically) meaning the #1 website in the world, according to Quantcast, is #1.
Alexa Rank focuses on telling the popularity of a particular website. It can show the details about the ups and downs in the pageviews and visitors demographic segments. Like Quantcast, Alexa combines their data into a list of most popular to least popular websites in the world. In the case of Alexa, they rank the top 1 million websites.
These two metrics are very similar in what they measure. The numbers are different because they use different algorithms.