core-js is the more common of the two in our index: it runs on 17,336 websites against 20 for JsRender, and 11 sites use both. Counts come from real scans, not vendor claims.
core-js is a modular standard library for JavaScript, with polyfills for cutting-edge ECMAScript features.
JsRender is the template library. The library is developed and maintained by Microsoft employee Boris Moore and is used in projects such as Outlook.com and Windows Azure.
11 sites run both core-js and JsRender. That is 55% of the sites using JsRender, the smaller of the two.
Want to know which one a particular site uses? Look it up and see its whole stack, or compare two sites side by side.