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/
Plus I remember at one point Obama was talking about duplicate agencies/programs and consolidating them. Then Trump talks about "cutting the fat", but I do see how being separate in this case actually seems like it helps.
I think the press articles that sometimes lump it all together can be frustrating at times and I like Matt's descriptions above. Another important difference: when USDS attends a meeting it's the "White House" in the room, whereas 18F sitting in GSA (which does other things like support government real estate) gives us a different context.
We both, though, benefit immensely from strong civil servants that partner with us. There's a narrative out there that "silicon valley" people have come to DC to "rescue" all of the government, when many people in both orgs have government experience (or come from the non-profit sector). Our work is impossible without partners throughout government who know a policy space and are ready to benefit from different tools and processes, and I wish those folks (or even just the tools and processes such as http://www.gao.gov/products/GAO-12-681) got more attention than the "hero narrative."