If your assumption is correct that they are paying less because of a statutory limit on how much property taxes can increase each year, then the fact that you, as a recent buyer, are paying more for a house with the same assessed value but that was bought many years ago is exactly what the law is supposed to do. In that case you have zero grounds for any sort of appeal.
The idea of such laws is that if someone bought a house, say, 10 years ago for $100,000, and then there was a market boom in the area and the house is now worth $500,000, it's unfair in some sense to make him pay taxes based on the $500,000. He didn't know values would skyrocket. He may well have bought the house after carefully considering what the property taxes would be, and concluding that he could just barely afford them. To say that now, because of factors outside his control and quite possibly outside the ability of anyone to realistically predict, his taxes should skyrocket so that he can no longer afford them and is forced to sell his house, seems unfair to many.
But for someone who bought a similar house next door for $500,000, he knew what the taxes were and made the decision to buy with that knowledge. (Or if he didn't check, the information was readily available and it is his own fault.)
Of course you can argue that it's unfair that you have to pay higher taxes for a house with the same market value. You could start a political campaign to change the law. But as long as the law is what it is, you are not going to win any appeal. It would be like going to court and arguing that it's unfair that you have to pay income higher taxes than someone with the exact same income but who has more children because the other person gets more personal exemptions. Or arguing that it's unfair that you have to pay gasoline taxes every time you fill your car's tank while people who ride the bus don't. Etc.
Sometimes a judge will buy your argument that a law is unjust and declare the law unconstitutional, but I think that's a very long shot on a case like this. And frankly, if you won, the likely result would be that your taxes would be exactly the same while your neighbors' taxes would all go up. I doubt this would make you popular with the neighbors. And you would see your future property taxes rise faster, so you would be shooting yourself in the foot for the satisfaction of knowing that you also shot all your neighbors in the feet.