Start here. This is the direct spoken answer to practice first.
Overview
The main difference is not whether a model is open or closed; it is who owns the serving, capacity, security, upgrades, and failure recovery.
A hosted API lets a provider operate the model and serving infrastructure, which usually gives faster adoption, elastic capacity, and access to managed safety and observability features. Self-hosting an open-weight model gives more control over deployment, data path, model version, and customization, but the team owns hardware or cloud capacity, inference software, scaling, patching, monitoring, and availability. Neither option is automatically more private, cheaper, or more reliable; the implementation and workload decide that.