There aren't any laws yet. There are only FCC policies that are being enforced, and companies choosing to comply with them or not.
The reason you're getting conflicting responses as to whether it constitutes censorship, is because there are some very good arguments for net neutrality being a way to protect consumers from censorship, and some very good arguments for net neutrality itself being a form of censorship.
Network Neutrality is a principle. It says that the people providing you internet shouldn't have a say in what you do with it (if you're committing crimes, that's a different issue). The idea is that a content provider can't pay money to a service provider to get more bandwidth dedicated to their content.
The censorship issue from the user's end is that a content provider shouldn't be looking at where you are getting content from to begin with (possibly unless there are law enforcement issues; again, this is a separate issue).
To put it simply: the post office shouldn't open and read your mail.
There are also a lot of technologies that allow people to communicate things in secret to other people. Things like bank account passwords, business info for people that work over the internet, personally identifying information, and the rest. Lack of net neutrality would mean that all these technologies would get a lower bandwidth. This means that if you have a job securely working on a remote computer, it may become literally impossible for you to do your job without net neutrality.
The other issue is that service providers could filter out information that isn't friendly to them. For example, Google could make it take an hour for you to load up a page saying bad things about Google. Politicians could similarly pay service providers to do the same thing.
The censorship issue from the other direction is because any law that affects net neutrality, takes some of the freedom service providers have to decide how to distribute internet to people. This can cause real problems and not just for the business involved. If a service provider was trying to get service to a hard-to-reach rural community, and was legally required to promote every kind of internet content equally, then as new services came out and the money simply wasn't there to keep up with it across the board, those rural communities would lose all their internet service. There are also ethical considerations as to how much of a role the government should play in making decisions for private business. The same considerations exist any time the government has to regulate just about anything. They are never easy, and a solution that makes everyone happy usually is never reached.
The happy medium is what we're already doing. Like news media, laws can only go so far in protecting the public interest. After a certain point, news broadcasters just have to be dedicated to the principle of reporting accurately and honestly, and likewise net neutrality is just something that service providers are going to have to believe in. That's not easy when the rewards for going against net neutrality are many, and especially not when people setting up cable and wireless networks view the networks themselves as an investment.
Recently, 5 attempts at passing Net Neutrality laws were made, and Google and Verison began a tiered internet program, effectively breaking net neutrality. So even though the happy medium has worked in the past, tensions on both sides are reaching a critical point and people are taking matters into their own hands. We are approaching a point where we as a society will have to decide what the lesser of two evils is.