Definitions used by this question:

• 'functional' Try is what https://github.com/lambdista/try is. Success and Failure are subclasses of Try. Exceptions aren't thrown.
• 'rx' Try is what https://github.com/aol/cyclops/wiki/Try-:-functional-exception-handling-for-Java-8 is. Exceptions are thrown.

Under what scenarios would one style be more beneficial than the other?

Of course, if the called function throws an exception, it's easier to use the 'rx' style. OTOH, it's still possible to wrap that function with another function that follows the 'functional' style.

Definitions used by this question:

• 'functional' Try is what https://github.com/lambdista/try is. Success and Failure are subclasses of Try. Exceptions aren't thrown by the called function.
• 'rx' Try is what https://github.com/aol/cyclops/wiki/Try-:-functional-exception-handling-for-Java-8 is. Exceptions are thrown by the called function.

Under what scenarios would one style be more beneficial than the other?

Of course, if the called function throws an exception, it's easier to use the 'rx' style. OTOH, it's still possible to wrap that function with another function that follows the 'functional' style.