Well I've only done just over 100 miles in her so I can't offer much of a report, but first impressions:
What's better on the RX:
1) The headlamps are a million times better than the HIDs on the LS. They have to be seen to be believed. Also, they turn with the steering.
2) The steering and seat motors are much quieter.
3) The doors "clump" better than the LS, which always "clouted" on mine.
4) The CVT transmission offers much, much smoother acceleration
5) Stereo and satnav are much better, although oddly enough, medium wave seems quite "dull". The CD player is a league above the LS, which always sounded harsh to my ears.
6) Surprisingly, for a 2 tonne car, it's bloody quick - it could easily keep up with the LS. I think I've already embarrassed a few diesel "faster than a Ferrari" Audis...
7) The switchgear around the steering column is much, much nicer than the LS.
8) Fuel economy - I'm getting mid 30s on the motorway. That's a smaller engine with similar performance in a brick-shaped body. Impressive, although the electric gubbins are a big factor.
9) Power boot. Opens and shuts from the key, lovely feature.
10) Folding door mirrors although sadly, not automatic when locking.
11) No external telescopic aerial.
12) Bluetooth and stuff (although not bluetooth audio and no aux input)
13) Better steering, more feedback
14) It's nice that the engine turns off as you approach the lights. And as you cruise around car parks. And when you start it early in the morning, it doesn't wake the neighbours - it just moves off, silently. That's a lovely feeling.
What's better on the LS:
1) The LS seats have more leg support and that heavily sprung base is very much missed on the RX.
2) The 4.0 engine is smoother and nicer to the ear than the 3.3 on the RX
3) I miss the coin tray near the ignition barrel (RX has a foldout cupholder there).
4) Audio system and satnav are better integrated
5) Spare wheel is more easily accessible (spare on the RX is under the car)
6) Better ride quality - the RX seems to bob around a little bit more than the LS, which just ploughs through bumps in the road.
7) Better brakes - the RX has regenerative braking and at least once I've had to apply more braking force than normal as it chooses between charging the battery and stopping the wheels. The LS is just "push pedal = slow down".
8) Smoother engine/transmission at low speeds - you can feel the RX's engine turn on and off sometimes, although perhaps mine would benefit from new engine mounts. Not a big deal.
The RX is definitely a much more sophisticated and refined car, but it has the benefit of being much newer. I like the higher driving position. The satnav system is hugely complicated though and will take me a long time to learn. It also has voice commands, but I doubt that'll work well.
Really though, what I'm most looking forward to is enjoying this marque's reliability. That's very important to someone who has to get up at all kinds of strange hours to go to work.