They also have different funding mechanisms that lead to different operating models. USDS funding comes out of a budget called ITOR and we can represent a source of "free" engineers for an agency. 18F operates on a cost-recoverable model where an agency is paying for help or services.
There are other, smaller differences too. For example, it's typically easier to work remotely for 18F than for USDS.
From a distance, the organizations both look like "tech in government," but I think they operate better separately. I have enormous respect for the folks in 18F, and the two orgs have collaborated on things like the US Web Design Standards: https://standards.usa.gov/