Spring has two two types of DI (Dependency Injection): setter DI and construction DI.
Constructor-based DI fixes the order in which the dependencies need to be injected. Setter based DI does not offer this.
Setter-based DI helps us to inject the dependency only when it is required, as opposed to requiring it at construction time.
I do not see any other significant differences, as both types of Spring DI provide the same features - both setter and constructor DI inject the dependency when the code starts up. Granted, constructor DI will do it through the constructor while setter DI will do it through a setter right after constructing the object, but it does not make any difference for the developer in terms of performance, etc. Both also offer means to specify the order of dependency injection as well.
I'm looking for a scenario where one provides a distinct advantage over the other or where one type is completely unusable.
When it comes to Spring specific pros and cons:
Constructor injection (from the definition) does not allow you to create circular dependencies between beans. This limitation is actually an advantage of constructor injection - Spring can resolve circular dependencies when setter injection is used without you even noticing.
On the other hand if you use constructor injection CGLIB is not able to create a proxy, forcing you to either use interface-based proxies or a dummy no-arg constructor. See: SPR-3150