A Lutz is a takeoff from an outside edge, not an inside one, else it's a "flutz" (a 0.1 deduction for a last-moment transition, a "serious error" for a longer inside edge before the takeoff).
It's among the hardest jumps because the rotation is counter to the curve of the takeoff edge. It's a toe-assisted jump, like the Toe Loop, but while that rotates in the direction of the curve (typically the right back outside edge curve, i.e. counterclockwise), the Lutz takes off, when done properly, from a left-turning curve described by the left back outside edge. (And, instead of the left toepick going in, the right one does.)
In that sense, the Lutz is more like (and in roughly the same category of difficulty as) the Toe Walley, which takes off on the other inside edge, but is "cheated" often enough that it is now officially treated similar to a Toe Loop in certain cases (mainly, because judges have a hard time telling one from the other, when the skater takes off on a flat, or nearly so).
I haven't done any of these jumps yet, having only recently started doing Waltz jumps in sequence (which is pretty cool, and has me swaggering around the rink like I just won Olympic Gold, but, hey ;-).
But my wife has had all sorts of struggles regaining, then losing, then regaining her Lutz.
And my impression is that the main challenge of the Lutz is to use the speed going into the jump against the toepick to change the direction of rotation of the skater to achieve the counter rotation (to go counter clockwise, after taking off on a clockwise curve).
That takes more of a pick-in and push-against effort than is required, as far as I can tell, of the other jumps. I mean, when I do a Waltz jump, I use the pick agains tthe ice to jump off and accelerate my existing counterclockwise rotation in the air, but with a Lutz you have to establish a rotation in the other direction. (And with a Toe Walley, instead of pushing against the ice with the pick, I guess you have to pull it towards you to change the rotation.)
So maybe the best Lutzes are those that don't just take off on an outside edge, but have true clockwise-rotation entrances requiring the skater to truly establish a counter-rotation as part of the takeoff. Otherwise all the skater needs to do is be flexible enough, after possibly establishing counter-clockwise rotation to make things easy, to do a bit of an outside edge before the takeoff to call it a "Lutz". That'd make me just fall over, hard, but maybe some skaters could make it work.
Anyway, the only difference between a Lutz and a Flip is that you take off on the inside edge of that (usually left) skate when you do a Flip, so you're already establishing the rotation on-ice that you'll be turning in while airborn.
Hence the term "Flutz" -- or "Flip-like Lutz" -- when a skater makes like they're going to do a Lutz, but instead switches to the inside edge just before takeoff. Makes things easier.
Oh, and at least a few judges would really love it if you Lutzers would do some between the blue lines for a change!! Seriously, you might get an extra .2, .3, or even .4, from what I've heard them saying.
Everybody wish me luck this Saturday; I'm supposed to be skating in a show that afternoon/evening, a show full of other students of a skating-fundamentals seminar, most of whom are much better than me, and I'm debuting my own choreographed program, complete with my new-found waltz jumps. I haven't skated in an exhibition or competition in over a year, and then only one of each!
but I feel pretty good in the company of a bunch of Olympic and World champions, who happen to be impressed with Sasha's spiral -- the most.
And just when I got at the end it ended up being about toe jumps. 
but Sarah belongs to that exlusive Olympic gold medalist club, so her opinion must count more than the judges, according to you. I thought Sarah said her spiral is the best #1.
