value of using React.forwardRef vs custom ref prop

Even React docs mention the custom ref prop as a more flexible approach to forwardRef:

If you use React 16.2 or lower, or if you need more flexibility than provided by ref forwarding, you can use this alternative approach and explicitly pass a ref as a differently named prop.

There is also a gist, in which Dan Abramov writes about its advantages:

• compatible with all React versions
• works for class and function components
• simplifies passing a ref to a nested component several layers deep

I would add, that passing refs as usual props does not cause breaking changes and is the way to go for multiple refs. The only advantages of forwardRef coming to my mind are:

• uniform access API for DOM nodes, functional and class components (you mentioned that)
• ref attribute does not bloat your props API, e.g. if you provide types with TypeScript

Does passing a custom prop affect diffing when it comes to rendering and cause additional renders?

A ref can potentially trigger a re-render if you pass an inline callback ref function down as prop. But it is a better idea anyway to define it as class instance method or via some memoization like useCallback.