They are both opposite sides of the same coin for me.
Graceful degradation is a web design technique, the website is designed in full scale with advanced UI (User Interface), features and enhancements as per the latest version of the web browser.
Then it slowly steps down with each step, only keeping the core functionality by the end. This way the direction of layers remains opposite for Progressive Enhancement and Graceful Degradation.
Progressive Enhancement is a technique of creating cross browser compatible web design wherein the highest priority while developing is keeping the core web page content, while the other complex add-ons and features remain secondary.
While the main idea behind the two methodologies is to establish a cross browser compatibility, where the functionality of the website remains intact on different browsers. The Progressive Enhancement methodology is implemented keeping the basic versions of the browsers in mind, on the other hand, Graceful Degradation focuses more on the advanced versions of web browser and slowly drops down enhancements to be compatible with the lower versions.
Also, since Progressive Enhancement methodology retains the core content in all the versions, it delivers good results for SEO(Search Engine Optimization) of the website, where the keywords could be detected in all the versions. This makes Progressive Enhancement a preferred technique in web development.
Irrespective of your pick out of Graceful Degradation and Progressive Enhancement there lies a common objective for adapting any of these 2 web development approaches i.e Cross Browser Compatibility.
In order to achieve the common goal, a detailed Cross Browser Testing is vital to ensure that your website or web app runs seamlessly across modern as well as legacy browsers before making the website go live.