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Perhaps the most important different is how they operate regarding deciding when to forward multicast traffic.  Dense mode uses a "flood and prune" model.  When a muticast source begins to transmit, it floods the stream everywhere.  It then checks whether any hosts on a particular flooded segment want the multicast stream.  If none do, it then prunes the stream.  While the multicast source continues to transmit, the flood and prune model is repeated.

Sparse mode uses a "pull" model.  When a multicast source begins to transmit, it's only forwards to segments that contain a host that requested that particular stream.

If most, if not all, network segments want a multicast stream, then dense mode works fine.

If few, if any, network segments want a multicast stream, then sparse mode is the better choice.