I had a 2003 LS 430 and swapped for a GS430 and now an RX so here is my 2p worth. It's comparing apples and Oranges really but here goes...
1) LS has lazy V8 - RX has "zingier" V6 - LS has quieter engine
2) Ride is more bouncy in RX - not a limo like LS but handles very well and is very comfortable.
3) RX not as big inside as LS but not that much smaller - does however have practicality of hatch and fold down seats to make a BIG boot.
4) My 350 gets around 20 mpg around town and 27 ish on Motorway - actually through roadworks with cameras on M1 I have seen 30+ a few times. Be warned that compared to LS it is a small tank (72) litres so range is about 320 miles.
5) Spec - I have SE-L which has air suspension, adaptive headlamps, Makr Levinson stereo, ICE, cruise control, Sat Nav, leather, electric everything basically, reversing camera (I also added front and rear sensors) so you will pretty much have every toy you will need.
6) 4WD obviously - works all the time in the background. LS was very stable car but like all RWD you could slip the wheels in wet etc. RX doesn't do this. You can blast out of wet side roads/junctions with confidence!
7) RX is shorter than LS - much easier to park
Overall in a perfect world that didn't involve parking in multi storey car parks I would pick an LS 430 however I can say that my RX is a fantastic car and I have no regrets about buying one.