The difference for passenger cars is pretty small, but it seems to be quite real. The less you maintain your tires the bigger the effect is. In large fleets of vehicles, the effect can be quite large.
Further investigation of the site justinm410 linked to, getnitrogen.org, shows a number of studies that (assuming they're all legit) show some real difference between nitrogen-rich and regular air.
With passenger vehicles, the fuel savings were between 2 and 8% depending on the study and driver behavior. A driver who regularly maintains their tire pressure sees much lower benefits than one who just gets the pressures checked during oil changes (about twice as much fuel savings if you only occasionally check pressures). One could argue that just checking your tires more often would save as much as nitrogen-filled tires, but the reality is that most people don't, so a public campaign to use nitrogen for tires could potentially see significant savings across a population.
The biggest differences are seen on large fleets of heavy vehicles, where fuel savings were between 6 and 23%, and tire lifespan was increased anywhere between 40 and 86%. It seems that fleet vehicles aren't maintained as well as passenger vehicles, plus they have to survive most more strenuous conditions.
There are two main causes for the savings. First, nitrogen-filled tires leak more slowly, so your tires will be closer to their correct pressures at any given point. I.e., if your tires are normally at 32 psi, the regular air might leak to 30 psi in one month, while the nitrogen is still at 31.5 psi. Second, nitrogen-filled tires see less variance in pressure during use. This lowered variance is mostly attributable to the lowered moisture, so a device to dry normal air might see similar results.
There's also the issue of safety. A properly-filled tire will perform better, and with more predictability, than an improperly-filled tire. If the tire maintains a more consistent pressure, it follows that it will be somewhat safer, but there's no data on whether that difference is significant. Additionally, at least one of the linked studies cites "decreased number of blowouts" from nitrogen compared to regular air, but there's no information about how much of a decrease they saw.
I don't have any way to verify the studies, but they've got papers from Bridgestone, Firestone, the Drexan Corporation on behalf of the Canadian government, among others. Plus Jay Leno says nitrogen is cool, and he's a celebrity, so obviously you have to listen to him. I tend to take a site's source material with a dose of salt until outside references verify it. But at face value, it seems to indicate the sign in the OP is basically accurate.
Edmunds.com investigates these studies, and concludes that while the studies are basically accurate, it's probably more efficient to just buy better tire-measuring equipment and get on it more often. Still that doesn't really mean anything for people who know they don't ever check the tires. Edmunds themselves did a study showing even "car people" really aren't as good about this as they should be.
Many people, even at an automotive Web site like Edmunds.com, have never checked their tire pressure. This means the pressure is only adjusted when the tires are rotated, about once every six months to a year. This isn't often enough. Tires should be checked and adjusted at least once a month.
We also found that many drivers don't know what level to fill their tires to. Many people thought the pressure was listed on the sidewall of their tires. This is wrong. The correct tire pressure is listed on a placard found in the car's door, doorjamb, glove compartment or owner's manual.
Still more people seemed to be simply guessing how much air to put in their tires. They pumped air into their tires now and then to make sure they were over the required amount. But they didn't check and adjust the tires to make sure the tires were all approximately at the same level
The Tire Rack
The TireRack.com does their own study, where they conclude:
Overall, inflating tires with nitrogen won't hurt them and may provide some minimal benefits.
Is it worth it? If you go someplace that provides free nitrogen with new tires, why not? Additionally we've seen some service providers offering reasonable prices of about $5 per tire (including periodic adjustments for the life of the tire) to a less reasonable $10 per tire (with additional costs for subsequent pressure adjustments) or more as part of a service contract, which we believe exceeds the value of nitrogen's benefit.
Rather than pay extra for nitrogen, most drivers would be better off buying an accurate tire pressure gauge and checking and adjusting their tire pressures regularly.
Note that, again, their bottom line depends on drivers paying more attention to their cars. This is solid advice, but only helps if drivers follow suit. Otherwise, it seems nitrogen may be measurably better if it's cheap or free. Of note, both Get Nitrogen and Edmunds estimate about $100 a year could be saved with more consistent inflation. This means you'd still save some money, even with relatively more costly fill-ups.
Response for the specific claims made
- Take the lead! Ride smart! Marketing hype, of course, but potentially accurate.
- [Nitrogen] for your SAFETY & ECONOMY Presumably safety is true, but it's unknown how true (even Edmund's other study isn't helpful here); the studies show economy is true, to some degree.
- Nitrogen molecules are bigger than normal air / Less leakage This paper explains that oxygen molecules have a slightly smaller "kinetic diameter", meaning they permeate through the rubber slightly faster than the nitrogen molecule. Since most of the non-nitrogen part of normal air is oxygen, this statement is true, if somewhat imprecise.
- Fuel effecient [sic] The studies on the site show this is true, citing from 2 to 23% better fuel economy.
- Less irregular wear It stands to reason less variance in tire pressures would lead to more even wear on the tire, but it's unknown if this is significant.
- No presence of Oxygen / No oxidation These are somewhat hyperbolic, but true according to the site's cites.
- Longer tyre life Shown by the studies to be true, citing 31 to 86% longer tire life, but these high numbers are for people who never check tire pressures and/or fleet vehicles.
- No deterioration Again hyperbolic, but possibly true if the "decreased number of blowouts" is significant, and is caused by deterioration rather than low pressure.
- It's no wonder why aircraft and formula 1 cars Use nitrogen to inflate their tyres! It's probably a bit silly to compare typical passenger cars to heavy duty equipment or purpose-built race cars, but the basic premise behind the use of nitrogen in these fields still holds in the field of passenger cars, albeit to a much lower extent.
The sign is accurate, but the real-world effect is mostly very small for passenger vehicles in normal usage. If you're lazy about maintaining your tires, the real-world effect of nitrogen becomes more practically significant, though still small. Safety is some concern, but a difference of 1 to 2 psi isn't likely to be the difference between no crash and seventeen fatalities.