gateway, middleware

Well, it's kind of vague terms depending on the field you use (network, platform, etc.).

If you look at a concept that's similarly accepted,

Gateway - Access to Network Concept

Middleware - Approach to Platform Concept

It seems ambiguous to use the roles of these two terms separately. In terms of interoperability, the goals (objectives) pursued are similar, but it is difficult to say that they are similar in a large framework.