I did forgot to mention in my overview of the original "Little Fuzzy" by H. Beam Piper that the characters (or at least the protagonists) tended towards the Heinlein-esque uber-competent design, firmly planted on the straight and narrow moral path (at least for an old west view of morality) but still eminently likable. In Scalzi's version they're definitely more realistic in just about every aspect. Less super-competent, less super-moral, and in some cases less likable as well. Also, there are a few more women now (and they're handled better than in the original.)
The infallible truth machine (which was a key plot component) and the anti-gravity generator (which wasn't) were both dropped, while space elevators were added. (It will be amusing to see in another 40-50 years if space elevators have become a reality, or if they'll seem as quaint as anti-gravity and infallible truth machines usually do now.) The main upshot of all that of course is that without the "veridicator" the court room scenes involved a lot more lying and prevarication and a lot more need to provide evidence to support the claims being made.
The planet they're on is much more "frontier-sy" now. I forget the population in Little Fuzzy, but they had at least one real city and i think over a million people overall. In Fuzzy Nation the entire planet has about 100,000 people and the capital is something that seems to barely qualify as a town, with a population of a few tens of thousands i believe.
The other big change that comes up right at the start is the idea of protecting the environment being a consideration now. In the first book the same people who thought the Fuzzies were worth protecting thought nothing of pretty much just strip mining the planet for resources. It was kind of an amusing dichotomy. Of course there are still people in the new book who would be happy to strip mine everything, but there are also people who disagree and government regulations about exactly what they can get away with.
So now the somewhat spoilery part...
The mega-corporation, or at least the people heading it up on the local planet, is still the bad guy, but it's no longer quite as malevolent. They're certainly willing to do bad stuff, but they no longer seem to be doing it with quite so much glee. They're certainly not adverse to harming the Fuzzies, but they don't seem to be going out of their way to organize genocide either. Vis a vis the more modern take on things it's pointed out that you don't need some giant conspiracy for something like that to happen. If you've got a hundred thousand people or so, all of whom will be out of a job if the Fuzzies stick around, you don't really need direction from the top in order for things to get ugly; cf. the traditional human treatment of wolves, or what might have happened with the spotted owl if the relevant forests hadn't been right next to (relatively speaking) a highly developed area with lots of law enforcement around.
I'm guessing perhaps that was the reason for shrinking the population? More logger/miner/farmer types whose livelihoods would be threatened and fewer insulated "townies" to get sentimental about the Fuzzies and try to protect them.
Although there is one notable bit where the new corporation is notable more "evil." In the original book at least some of the members of the corporation deluded themselves into thinking the Fuzzies weren't sapient but then had sudden changes of hearts once it was proven otherwise. In the updated version you get the sense that they really don't care one way or the other about their actual sapience, either before or after it's proven. They just want the Fuzzies out of their way.
The government in the new book is still fair and willing to bring the drop the hammer when provided with proper evidence in a court setting, but is no longer a crusading paladin out to take the corporation down like in the original version.
Holloway (the main protagonist) is a much more morally ambiguous character. It's not even entirely clear at first exactly how much he's going to side with the corporation and how much he's going to side with the Fuzzies. You keep thinking he's got to come around since he's the protagonist (especially if you read the original book) but he does drag his feet a bit and make you wonder.
So overall the general plot is similar but the details and characters are very different. As opposed to say Star Trek (which often gets brought up in comparison) where the general plot was completely different but they mostly tried to stick to the original characterizations and details (at least pre-divergence.)
It's hard to say which version is better because they're definitely very different from each other in the details, but i enjoyed both very much and in both cases couldn't resist burning through them much faster than i usually allow myself to do. However I'd certainly recommend Scalzi's version for anyone who has a problem with the "quaint" ideas of characterization and future tech that were prevalent in the golden age of SF.