Per the doc, loose WeakReferences must be cleared by a running GC.
Per the doc, loose SoftReferences must be cleared before OOM is thrown.
That's the only real difference. Everything else is not part of the contract. (I'll assume the latest docs are contractual.)
SoftReferences are useful. Memory-sensitive caches use SoftReferences, not WeakReferences.
weak_ref.get(). When it's null, you learn that between this duration, the GC ran.
As for incorrect use of WeakReference, the list is endless:
a lousy hack to implement priority-2 softreference such that you don't have to write one, yet it doesn't work as expected because the cache would be cleared on every GC run, even when there is spare memory. See https://stackoverflow.com/a/3243242/632951 for phails. (Besides, what if you need more than 2 levels of cache priority? You'd still gotta need a real library for it.)
a lousy hack to associate data with an object of an existing class, yet it creates a memory leak (OutOfMemoryError) when your GC decides to take a break after your weakreferences are created. Besides, it's beyond ugly: A better approach is to use tuples.
a lousy hack to associate data with an object of an existing class, where the class has the nerve to make itself non-subclassable, and is used in an existing function code which you need to call. In such a case, the proper solution is to either edit the class and make it subclassable, or edit the function and make it take an interface instead of a class, or use an alternative function.