When I was developing my Core Web Vitals extension, I realized the difference and now always use only local metrics at the very beginning of development.
Google Search Console uses real user data (Field Data) from the Chrome User Experience Report (CrUX), which collects metrics from users in real-world conditions. These metrics are highly averaged, as they take into account data from users with different devices and internet speeds. Additionally, they are aggregated by groups of URLs rather than individual pages.
PageSpeed Insights displays both Lab Data (e.g., slow 4G and a mid-performance device) and Field Data. Lab Data is measured in a controlled environment and may not reflect real user conditions.
Your local data, which you can check in the extension (Core Web Vitals Test) or directly in the browser under the Performance tab, is calculated specifically for each page you visit. These local measurements also contribute to the data used in CrUX.