Hey - you two are saying essentially the same thing. It all "hinges" on the definition of a takedown.
Mike began the thread by asking for definitions/differences. He then followed Rory's lead and broadly classified anything that puts the opponent on the ground as a takedown - meaning that a throw is a type of takedown. This is true but forms the basis for the confusion that follows.
Mike comes from a Japanese ju-jitsu and judo perspective. So does Fred and Karateka. In judo there is ground fighting similar to BJJ - but only after a controlled throw for additional points if the throw wasn't ruled an ippon (is this correct - working from memory?).
The ju-jitsu that Mike teaches focuses on throws (similar/same to judo) and control of the uke after the throw - not so much on groundfighting although it is also covered. I believe that this is the same for Karateka.
How about pondering this: "Are all throws takedowns?" contrasted with "Are all takedowns throws?" I believe that the former is true while the latter is false. I also agree with Karateka - and think that Mike, Rory and Fred may as well - that on the battlefield a nicely executed naga waza has more utility than a perfect double-leg takedown to the mount position - especially if there's more than one opponent. But Mike cautions about getting too focused on one technique - adding that he'd use whatever technique seemed right for the situation.
The missunderstanding is thus based upon the broader definition of takedown to include throws - Mike and Rory accept this, Fred and Karateka have not bought into this definition. I agree that all throws are takedowns, but suggest using the terms in the sense that Fred suggests - takedowns are more from a grappling (BJJ) perspective for one-on-one grappling arts and throws are more from a ju-jitsu/judo perspective.
Finally, having had my a** handed to me by Mike more than once on the mat (I'm a grappler), I'd rather be taken down by him than thrown by him. Something about gravity and landing correctly that I haven't figured out!