Hey @dfisher while I was writing my answer, I saw that @diegop posted his. I took a little bit a different approach and tried to explain the basic task of both the API Boundary Node and the HTTP Gateway:
To answer your question, let me first focus on the replica nodes that host the actual canisters and then bring the API Boundary Nodes and the HTTP Gateways in the picture.
Replicas expose an API to interact with the canisters that they host. Think of a simple counter canister, for example, which allows to obtain the current value (query) and increase the count (call) through the API.
If you wanted to make an API call to increase the count, you would have to first figure out in which subnet the canister is running. Then, you would have to find all the replicas that are part of this subnet. Only then, you could submit the API call to one of the replicas. This process is called routing.
The API Boundary Nodes take care of that for you. You can think of the API Boundary Nodes as forming the edge of the Internet Computer. You can submit your API call to any of these API Boundary Nodes and they will make sure the right replica node inside the Internet Computer receives it.
API calls are specific to the Internet Computer and only applications built specifically for the Internet Computer support them. Our goal, however, is to make the canisters accessible for anyone using their standard browsers, which only “speak” HTTP(S).
That’s where the HTTP gateway enters the picture. It provides an HTTP endpoint that acts as a gateway to the Internet Computer by translating the HTTP requests from the browsers into API calls.
To sum up: the API Boundary Nodes handle all API calls to make sure they arrive at the right replica, while the HTTP gateway translates the HTTP requests from the users’ browsers to API calls that the replicas can process.
Note: I tried to explain the difference at a very high-level and therefore simplified and left out a lot of details. If you want to know more, check out the Boundary Node wiki page and the forum post on the future Boundary Node architecture. Let me know if you have any questions or I didn’t explain something properly.