Good question.
Our migration was fairly direct. We did it in around a week with some cleanup in the following week. There were analogous conventions throughout just about the entire migration, so they're very similar.
One thing we liked was better TypeScript support. We're big on types and proper support, and most of our company is based on TypeScript and Rust. Gatsby can use TS, but you need to set it up and we came across the odd hiccup in the tooling. Next supports this off the shelf and so far, it shows. We haven't had any issues with compilation anywhere in the pipeline or other tooling.
The way Next handles dynamic pages is different, and in my opinion nicer. It also supports something they call incremental static generation which allows pages to be served statically, but incrementally updated as dependent data changes. For example, you know the team in the lead will be the same for thousands of requests, so you might as well serve that statically until the leading team changes. Next will manage that content update for you while serving the content statically for as long as possible. When we migrated, Gatsby didn't support this feature.
Next handles pages differently in general. It maps your directory structure under /src/pages, so that /src/pages/hello-world.tsx will be yourwebsite.com/hello-world. It allows for dynamic content using a convention in which file names can represent a variable, where src/pages/foo/[bar].tsx will make it so [bar] is a variable in the query intercepted by getServerSideProps when you visit yourwebsite.com/foo/whatever.
Otherwise I think the main snag we had with Gatsby was that the team was slower to address issues, introduced bugs more frequently, and we began to see that it was impeding our progress. Our software was working independently of Gatsby and wasn't the issue, and no matter how closely we worked with the maintainers, we couldn't move the needle on any blocking issues. With Next it has been a much different story. Issues are unblocked routinely and the maintainers seem much more engaged and productive.
There's more, but those are the main differences that come to mind.