Take what I say with a grain of salt since I haven't used either. The API folder in Next.js simply exposes API routes that you can use in the frontend code of your Next.js app, or not use it if you don't want to.
Remix does support this behavior, but it also supports exporting a data loader function and/or an action function for forms that can only consumed by by the page you're working on. All of that runs on the backend. Again, very similar to how things are done in PHP where data is loaded from a DB and then consumed all within the same file.
Their form demo is definitely worth checking out. You could build the same thing with a Next API route. You’d need to write some code to do the validation that they do, but it should largely look something like their hook does. Hopefully Next will build something like it because this is a much more sane way to write forms than the controlled inputs and state way.
I do think it’s nice having it in the same file as the form itself, but it’s not clear how well that’ll scale in a more complex page with many forms. So I think Next could get most of the way there with a few helpers to make a form submit to an API route with similar DX.