I think you are referring to API vs 1-click embedding? Actually, both approaches are using iframe to embed. dashboard.
1-click embedding is more straightforward and it was designed for non-technical user to embed dashboard in internal web app such as wiki, sharepoint. Or they may embed dashboard in public website. If you are embedding in internal web app, you need to grant dashboard access for user. Users will be prompted to sign-in or you may enable SSO. If you embed in public website, everyone in internet can access and you cannot apply Row Level Security(RLS)
with API embedding, you can enjoy more functionalities such as RLS in both internal web app and public website. You may use namespace to isolate user group for multi-tenancy. And you can empower user to author dashboard