To my understanding , fragmentation will be eliminated in both the ways, Rebuild of index will use the existing index and hence it is much faster and needs much space on the disk.

well dropping is also a good thing, just like starting from fresh , but its a time consuming process, sometimes u are not affordable to do it.

Rebuild with NOLOGGING (no REDO will be generated ) is much faster way of creating the index, incase of failure of Index creation , u always will have the old, existing one.