I thought that, at first, they do have somewhat of a new society. By getting rid of Jones, they controlled all the means of production: they had to contend with maintaining their capital - which caused problems. Jones had been the provider for that capital, but the animals felt they could do just as well themselves. Until the feud between Snowball and Napoleon and until the 1st windmill disaster, the animals seemed to be doing alright as a commune: raising their own crops, and essentially taking care of themselves. But the pigs created a division and got greedy, and started to see that this greed would rely on outside capital and harsher working conditions for the animals. This meant alliances with other corrupt farmers (Frederick) and basic enslavement of the animals who started what was once a new society. I've read that Frederick might symbolize Hitler and Napoleon symbolized Stalin (and Napoleon obviously). I always thought this was a satire of Marxism, Feudalism, Capitalism and failed Socialism (Russia). Since Orwell favored some kind of Democratic Socialism, this might be true.
In any case, this story is a satire of the Russian Revolution. Orwell saw this revolution as a disappointing failure: disappointing because what could have been a state where the workers shared the wealth became - like Animal Farm - a totalitarian regime where its people were slaughtered. There is also a reference to the French Revolution. Many authors, artists were also disappointed in its outcome, because, like the Russian Revolution, an old owning class was simply replaced by a new one. So, in the end, like the previous poster said, only personnel changed: things were essentially the same or worse.
At the time of the rebellion against Farmer Jones, the animals think they are building a new society, but in a real sense, they are not.
The only things that really change are cosmetic. Other than, I guess, the fact that no one gets killed for food.) There is a new leadership which is (at first) going around on four legs instead of two. There are slogans touting the equality and brotherhood of animals. And the animals are to some extend directing themselves.
But these changes don't really matter much. Most of the animals are still oppressed by those above them. So I don't think it truly is a new society.