As I look at the ISTE Standards I see a strong parallel with the outer arch of the Framework for 21st Century Learning graphic while the Common Core Standards can be found in the inner arch and also on the top support. "The Framework presents a holistic view of 21st century teaching and learning that combines a discrete focus on 21st century student outcomes (a blending of specific skills, content knowledge, expertise and literacies) with innovative support systems to help students master the multi-dimensional abilities required of them in the 21st century and beyond" (P21 website). This shows that the two are interrelated and must be used together in the classroom, along with the supports, to provide the best possible education for students today.
Both sets of standards claim to have very similar goals:
"The CCSS were created to ensure all students graduate from high school with the skills and knowledge to succeed in college, career and life" (http://www.corestandards.org/about-the-standards/) while the ISTE Standards S have been defined as "the standards for evaluating the skills and knowledge students need to learn effectively and live productively in an increasingly global and digital world" (http://www.iste.org/standards). I see the two sets of standards working together to provide students with the best possible education in today's world.