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Or put another way, All Flies Are Lures, But Not All Lures Are Flies.

Actually, there may be a 'legal' difference and depend on what state you are in and specifically where you are fishing. Aside from that there are some things that are not the same as other things and...

A 'lure' and a 'fly'.
Didn't want to hijack another thread and this subject probably deserves a thread of its own. Without getting into a heated discussion let's see if we can come to an agreement...