I don't. I used them first over 15 years ago when Haskell was "the next big thing" and I learned it at university. Fact of the matter is that not enough code has been written with them to say that we know how effective they are.
> If what you said was true, it wouldn't make sense for a hybrid language like Scala to have both. Obviously Scala's designers don't consider exceptions and Either/Maybe "identical", do they?
I wouldn't take lessons from Scala. Scala has inheritance types, sum types, structural types and dynamic types, each can be (and is) used in isolation to solve almost all problems. If there's any language with lots of overlapping features, that would be Scala. After all, it's a language designed to study various approaches, not to recommend the "best" ones.
> You can silence checked exceptions with reckless abandon, but you cannot silence a Maybe just as easily, without resorting to obviously frowned-upon practices
You can silence checked exceptions by doing exactly what you'd be doing to handle Maybe: You need to explicitly handle it. The difference is that you can catch multiple exception types in one block.
> For example, checked exceptions are terrible for generic code, composable code.
They're not. They just don't work too well with Java's limited effect system of checked exceptions. As a matter of fact, "checked exceptions" are the basis for effect systems in languages like Koka, and are also the basis of transformations of Haskell-ish monads to type-safe nested continuations, which are more appropriate than monads (yet equally expressive and powerful) for imperative languages.
Moreover, a system of checked exceptions composes much better than monads do. And you don't have to believe me. Phil Wadler says it[1] as does Oleg Kiselyov (who's trying to port the same idea back to monadic notation with free monads). This is my own treatment of the subject (which shows how continuations, of which exceptions are a special case, can do monadic composition without monad transformers, and provide type safety through checked exceptions): http://blog.paralleluniverse.co/2015/08/07/scoped-continuati...
> I truly don't see how you can seriously consider them "identical".
It's not a question of consideration. Wadler and Filinsky proved them to be the same (each from a different direction). See the link to my post above for the exact references.
Still, that's not to say that checked exceptions or Maybe are good approaches. We really don't have data on the subject. It basically comes down to personal preference.