I loved my LR4, it's an amazing vehicle and serves so many purposes. And if you really need the space, the RRS will feel/seem smaller. I purchased a MY13 RRS. It included the Silver Package which pretty much gives you every option but the rear entertainment system. I truly love it! To say one is better than the other isn't fair though. Because again, both serve their purpose. The RRS does have a lower roof-line and the rear slopes down rather than squared off. Therefore, you have less headroom, and less upper space in the boot. BUT, the headroom is still very ample. As the LR4, has headroom that's stratospheric. The rear seats in the RRS are also tighter, but, I'm 5" 9" so my seat isn't further back like it would be for the 6-footers, leaving plenty of leg-room for my wife/and Son's baby seat. I hauled around my in-laws/Wife/Son (total of 5-people) and no one was uncomfortable and even commented a couple of times they liked it more than the LR4 I had.
The RRS also does not have the 3rd row, which I only used twice in the three-years of ownership. Another note, the rear boot area, load level, is just a few inches shorter than the LR4... so you can still load it up with plenty of stuff. You just can't load high items, as the LR4 has that really high/squared off rear space. I have not found this an issue, I still haul around bags of stone for my landscaping... plants, same crazy stuff I hauled around in my LR4. So no issues there whatsoever.
The LR4 cockpit is nice, and fits the style of the truck. It's an amazingly comfortable space as well... gives a real sense of space. The RRS is more ****-pit orientated, the seats are very sport-like... they hug you, and the dash, swoops down, and the center-console area is higher and is at arm/resting height which gives you that jet-fighter/****-pit experience. You can hold the shifter and shift without the need to reach for it... it's sweet. And although the RRS still has that commanding seat height feature, you don't feel like you're going to roll out through the front windscreen and onto the bonnet. You still get that commanding feel... but it's more direct and matches the view points of the interior/and bonnet... giving you a sense of control over the vehicle, which makes sense, since it's a more sport orientated driving experience.
I haven't spent much time in a LUX LR4, but the RRS interior is quite refined and luxurious. It is also extremely quiet at head level, since it has the acoustic glass windscreen and driver and passenger feature. It successfully drowns out unwanted frequencies that are common surrounds. What the acoustic glass doesn't do, is drown out the V8 and rumble that is completely different than the sound of the LR4. The drive train is visceral. You can feel the V8 rumble, and that V8 oscillation at idle you'd experience in say a newer muscle car. It's an amazing experience especially in sport mode. Since the overall feel is completely different than the LR4. And for the record, the LR4 shouldn't be this way, it's a more refined, luxury family vehicle, whereas the RRS IMO, is more bad-boy... heavy metal-esque. Out to cause some trouble.
Back to sport mode: The sport mode is very special and intoxicating. Although when not in sport mode the character of the RRS is, well, sporty... when it's in sport mode, it shapes up to be a more direct and encompassing experience. The throttle mapping is quicker, the sport shifting is frankly lightening quick, I'm really amazed at how fast it shifts and the shift sound that goes along with it.
Downshifting is delightful and addictive, because in sport mode, the exhaust flexes its muscles and becomes more throaty, produces a sound and welcome vibration throughout the floor, center console, and buttock area that cannot be explained other than it reminds me of my e90 M3. While you're experiencing this, you look around the cabin, the size/feel of the truck, and just smile... because it's so bad-ass!!! Everyone who has been in my RRS sport suddenly realizes this isn't anything like any other SUV they've been in.
Another realization that I had after I purchased the RRS is how "Jeep-like" it is. It is the size of pretty much any mid-sized SUV - Jeep Cherokee from the 90s size. So when you look around the cabin, you still feel you're driving something spacious, but not overwhelmed in size. When I drove my LR4 by myself, I felt very lonely and vulnerable. It was just too big for me - felt like I was in a warehouse. And my Wife and Son only get in the car on the weekends - so it just made me feel alone.
When I'm alone now, I can enjoy the RRS as being mine... not a forced product on my mind-set because I think I need more space. If I had a bigger family I likely wouldn't have purchased the RRS. My selfish need to meet my mindset would have been deferred until a later time. But again, I didn't need the LR4 space and really wanted something I can enjoy. So any comparison here is really about what the family needs and what the family can accept.
I hope this helps paint a better picture of both vehicles. Each one has their purpose and I wish I could have both. At one point, I did try and convince my wife to take the LR4... but she loves smaller cars hence her love affair for her little Merc. I couldn't even convince her to get the E-class... anything larger than a 3-series or C-class is considered "too big." Annoying IMO. And although she loves driving the RRS, because of its sporty handling characteristics, she still says it's really big.
My wife likes the LR4, but she has a Tahoe, which I hate, but her dad gave it to her and she was raised only buying American-made trucks - mostly big SUV and giant vans (lots of kids). Luckily she keeps the Tahoe in UT, so I don't have to look at it much - hahaha! 
There are a couple guys in the NCLR club with newer RRS (pretty much stock) and they are still highly capable - though, some areas they do avoid, but I tend to as well! Haha. One of these days you have to ride along with me, in my LR4 - heck you gotta see my new LR4!!!
