I'm not quite sure what your plan is on LighterLife, but can explain what I believe the differences are. From my understanding with LL you also get CBT, which you do not get on CD, however, you do get an individual counsellor who will usually be very supportive (there are no group sessions).
The supplements are similar to LL, though I believe there are more flavours on CD. On CD you get soup, porridge, shakes (powder form) Tetras (cartons of milkshake) and bars.
LL is also strictly no eating (I think), however, the CD has various stages that you can go through. SS is 3 or 4 CD meals a day
SS+ is 3 CD meals + a 200 cal meal (protein and green veg, very restricted)
SS+ the other option is 4 CD meals and some milk
810 3 CD meals and a small meal (from list of allowed foods) and milk
1000 / 1200 / 1500 I don't know much about
I personally looked at both, and went CD primarily for the cost it is only approx £42 compared to £66, and havenever regretted it.