How a 600 SS/RR is different than your Ninja
First things first: Suspension - The Ninja 250 suspension is very soft and can soak up a lot of road imperfections without transmitting them to the bike's controls. That means that the rider won't feel them and so won't need to react. A Ninja ZX6R or ZX6RR has an incredibly stiff suspension in comparison, which comes with A LOT of adjustability. If the suspension is incorrectly adjusted, that alone can make the bike react badly to road imperfections.
Then, the stiffer suspension transmits all of the imperfections to the controls, so a bump in the middle of a corner can possibly upset the 636 a lot more than it would have upset the 250. How many times have you heard of 636 riders who complain about the bikes hitting something mid-corner and having a tankslapper (mild or otherwise) start?
Let's talk about those tires. There are tons of different tire compounds available for the SS bikes. Having the wrong compound rubber doesn't make much of a difference in the middle of a hot summer day, but in the winter here in NYC, having a set of Super Corsas on your 636 is a BAD idea, since they won't ever warm up to a usable temperature. A set of sport touring tires, like Avon AV45/46 ST's would be a much better idea, since they warm up to a usable temp range even if it's 20 degrees.
Now, on to the brakes. SS bike brakes are pretty extreme creatures. They are designed to bring a bike travelling at speeds well above 120 mph to a much slower speed, safely, in as quick a time as possible, while also allowing for great feel, and allowing the bike to do it over and over again without fading. If you tried the same thing with the brakes on the Ninja 250, the pads and rotor would heat up a lot, the brake fluid would start to boil, and the brake lever would lose its feel.
Now, that may sound like a huge improvement, but in reality it isn't. The brakes were designed for speeds above 100 mph, to slow the bike down to around 60 mph. At 45 mph, they are OVERKILL, and in some cases (GSXR) twitchy. If you grab at the brake lever because someone pulled out of a parking spot (not as dangerous on a 250) the front tire will lock up, and down you will go. Lots of riders learn this quite quickly, usually the hard way.
The most important difference is the power the SS bikes make. Most of the modern 600cc SS bikes make around or above 100 hp. Lots of people think that you can just not twist the throttle and be perfectly safe. Sometimes that's just not the case.
Scenario: Take that stiffer, twitchy suspension, add in riding on cold race rubber while going through a corner, leaned over quite far, and then hit a bump you weren't expecting, which causes your throttle hand to jerk a bit. Well, if you were up high on the rpm band, say about 8000 or above, you've got about 75 hp on tap at the rear tire. One throttle blip can send the throttle above 10,000 rpm in an instant in the right gear, which means you now have 92 hp being transmitted to the rear tire.
What does that mean? That means your rear tire just broke loose, since it's too cold, and you just felt the rear of the bike step out. So, you cut the throttle to decrease the power going to the rear tire. Bad idea. The rear tire grabs traction, but the bike is pointed in the wrong direction. You panic some more, and apply more throttle. The bike starts squirming more. Eventually, the front tire lets go, and down you go. If you're lucky, it's a lowside and you were making a left turn. If you're unlucky, you were in a right curve and go sliding into oncoming traffic. If you're really unlucky, you get spit off the bike in a highside and come crashing down on your shoulder, breaking your collarbone.
So, the whole package is different. It's not just more power, but everything. Several steps up should be made, but not everyone has the patience to do so. Some people just want a sexy, plastic-coated bike. Hopefully they won't destroy the sexy bike before they learn how to control it properly.