The real answer IMO is how they (don't) integrate with generics, first-class functions, and the type system in general. If you try using a checked function inside a stream() call you'll know exactly what I mean. Yes, it's technically possible to make a "functional interface" with a fixed number of checked exceptions, but in practice it's a huge PITA and most functions just don't allow checked exceptions at all.

Compare to ATDs, where any generic function taking a T can also be used with Result<T> exactly the same. (Not a perfect comparison, but there are lots of other scenarios where ATDs just compose better)