I have both(00 DIISE7, 06LR3HSE). The way I choose which to drive is whether I want to drive the DII or let the LR3 drive me. The DII has more of a truck drive and ride. The LR3 is more of a mindless driving effort, like a car. You can corner faster without too much body roll in the LR3 but my DII is very quirky with corners. I am not talking about racing around curves, ex... 40 mph curve-DII 45 mph, the LR3 55 MPH.
As far as off-roading, stock for stock. The LR3 wins hands down. I have done the same trails, creeks and mud holes in both vehicles. The LR3 will configure drivetrain for you particular needs, again mindless driving. The DII will definately get you through the same stuff but will depend more so on the driver and not the terrain response system. I dont do any aggressive off roading in either vehicle, so I cant speak about modding and Moab.
Sounds like you will be getting the DII, which is a great vehicle. I like the LR3 for more different applications. I have an Audi TT for a daily driver, but while it was getting body work done(damn deer), I did rely on the trusty LR3 for the 40 mile commute. Good luck with your decision making