NASA contracted to launch the Curiosity rover on the Atlas V rocket, for example, so how is this different? Well, the Atlas V wasn't developed commercially, it was developed as part of the EELV program. This was a USAF program which sought to develop launch vehicles to carry military payloads into orbit in the future. Boeing and Lockheed Martin developed those launch vehicles (which became the Delta IV and Atlas V) but government had a lot of say on the design of the vehicles and ultimately paid for most of the development cost of the vehicles. This is different from SpaceX's example where they developed their own launcher from scratch to their own design, specs, and manufacturing and management processes, etc.
Also, if you look at other NASA launches in recent history the contrast is even stronger. NASA drove and funded development of the Delta II, the Shuttle, the Saturn V, etc, etc.
Additionally, operations for the spacecraft component, the Dragon, is being handled by SpaceX themselves and not by NASA, which is a new thing, especially in the realm of manned spaceflight.
NASA isn't just buying launches, nor are they just buying Dragon spacecraft and launches, they are buying cargo delivered to ISS.
Beyond that, the interesting bit I think is the difference in the contracts that SpaceX has, and the contracts that have traditionally been given to space contractors. Traditionally overruns would be covered by government funding, but in the case of SpaceX they eat their own overruns. The result of this seemingly minor change in funding is that they are far less "quasi-government" than space contractors traditionally have been.
Also, and this is big, frankly SpaceX is operated in a much more nimble and entrepreneurial fashion. The primes, while having lots of very smart and well-meaning people working for them, have BDC-itis. Musk wants to dramatically improve the world and the species, and put a colony on Mars. The leadership of the primes seem to just want to make money. Musk/SpaceX is more exciting and compelling for a lot of folks, plus they're more likely to lead to spin-off missions down the road.
Early rocket programs were military programs, rockets make excellent weapons, they can reach around the world and touch someone in less than 90 minutes. The military paid for, and managed, programs for putting payloads into orbit. Of course the military doesn't have a rocket factory, so it hires contractors to build rockets to its specifications, and to research questions the military wants answers too, Etc. NASA worked with the same contractors as part of the civilian space program to give scientists access to space.
In the military programs, function was important, cost was not. That is because the military has "cost plus" contracting, they say "we need a rocket that does X" and the contractor builds it and charges the cost of building it plus the agreed upon 'profit' and the military pays it. There is no incentive to be economical and no incentive to change.
SpaceX on the other hand invested its own money in building its rockets (that's the 'private' part) and they are selling services on these rockets at a price that makes them a profit on their investment (that is the 'commercial' part). They have been re-using research done by NASA and the defense contractors in their work so they have an advantage over earlier efforts in terms of costs. That said, by approaching it as a business rather than a government contract they have removed a huge amount of cost out of the process.
Because of the process, SpaceX isn't necessarily constrained by the restrictions a nation-state might put on them (although in reality if they tried to sell launch services to say Iran the US would prevent that) They also have the right to re-use their technology for their own enterprise so if Bigelowe Aerospace needs someone to launch their space hotels, SpaceX can offer that service without either Bigelow or SpaceX needing 'permission' from the military.
The amazing bit here is that nation-state space programs were possible because nobody expected to 'make' money, they just took tax payer money and burned it at a high rate :-). In this case SpaceX is looking to recoup their development costs and then some. Not being under the whims of a nation-state budgeting authority (like the US Congress) allows them to make progress quickly.
Previous systems had heavy input for specs and design from NASA. With this, NASA basically says, "X pounds for Y dollars to ISS" and SpaceX is figuring out all the details.
Think of it like the difference between the cost of developing a new product such as an iPhone before scaling it out (cost plus) and the cost of the delivery of the final product (a final complete iPhone from the city store).