I believe the difference in weight is not negligible as you seem to mean: 2.35 kg for the 300mm II + 225g for the 1.4X III = 2.575Kg versus the 1.94Kg for the 400mm DO a difference of 635g.
If you have tried both combinations (with the same camera body) and you still think so I give up. I haven't - I do have tried 5DMkIII with a 500 f/4 IS II though - but I believe that in this category every weight saving counts.
In fact the change from a Manfrotto 055Prob + 410 Gear Head = 3.7Kg versus a Gitzo Systematic 3541LS + Markins M20 head = 2.3 Kg has made quite the difference for me (OK it's 1.4Kg less but you see the point... when walking everything counts)
Well, As I've already said, I have both and yes there is a difference as has been pointed out, the DO really is a pleasure to use, and it beats me as to why Canon don't push it more!...every time I go to a Canon show, you are struggling to find one in the line up, perhaps as it technically isn't an L lens.
If I had to get rid of one, it would have to be the DO, but only that it's the older lens and has, in theory, poorer IS.
However it won't go as once I get a second body (I sold my 7D a bit ago) I'll hang the DO on a crop body and use the 300 mk2 with or without extenders on the 1 Dx, if I'm travelling I don't like to constantly change lenses and bodies, especially in dusty environments.
I see that Canon recently filed a patent for a 600 DO...now that would be one beast of a lens..and perhaps I could lift it!!