One difference to something like email is that you can change search providers with minimal effort. There’s no server-side context to back up or migrate, no third parties involved: you just use a different URL.

When you get bad results from a search engine, you refine the query. Maybe try a different search engine! You have options. When it's your personal email address you don't have options. You are at the mercy of said email provider. You can run that email yourself, sure, but that isn't for everyone. Does that explain the difference?