What are the differences between shared pointers (such as boost::shared_ptr or the new std::shared_ptr) and garbage collection methods (such as those implemented in Java or
Those Shared Pointers (usually called reference counting) run the risk of cycles.
Garbage collection (Mark and Sweep) does not have this problem.
In a simple garbage-collected system, nobody will hold a direct pointer to any object; instead, code will hold references to table entries which point to objects on the heap. Each object on the heap will store its size (meaning all heap objects will form a singly-linked list) and a back-reference to the object in the object table which holds it (or at least used to).
When either the heap or the object table gets full, the system will set a "delete me" flag on every object in the table. It will examine every object it knows about and, if its "delete flag" was set, unset it and add all the objects it knows about to the list of objects to be examined. Once that is done, any object whose "delete me" flag is still set can be deleted.
Once that is done, the system will start at the beginning of the heap, take each object stored there, and see if its object reference still points to it. If so, it will copy that object to the beginning of the heap, or just past the end of the last copied object; otherwise the object will be skipped (and will likely be overwritten when other objects are copied).
The main difference lies, as you noted, in when the resource is released/destroyed.
One advantage where a GC might come in handy is if you have resources that take a long time to be released. For a short program lifetime, it might be nice to leave the resources dangling and have them cleaned up in the end. If resource limits are reached, then the GC can act to release some of them. Shared pointers, on the other hand, release their resources as soon as the reference count hits zero. This could be costly for frequent acquisition-release cycles of a resource with costly time requirements.
On the other hand, in some garbage collection implementations, garbage collection requires that the whole program pause its execution while memory is examined, moved around, and freed. There are smarter implementations, but none are perfect.