Out of the box, the new iPad looks just like the previous iPad. It's a hair thicker and a bit heavier, but other than that, it looks the same. The big change is the new Retina screen. The increased resolution is immediately noticeable compared to the iPad 2. Text is particular is noticeable sharp compared to previous generation. Presentations in Keynote look fantastic on the new iPad with no software updates or changes to the presentation itself. Since I deliver the majority of my customer presentations on an iPad instead of a laptop, this is a welcomed feature.
That said, most apps need to be updated to take full advantage of the new Retina display. Angry Birds, for example, looks exactly the same on the new iPad as on the previous generation. However, I updated a number Retina-enabled apps from Apple's app store and the difference is noticeable. I suspect we'll see a new Retina-enabled Angry Bird application soon (Angry Birds Space on March 22?)
In addition to the new screen, the rear-facing camera is greatly improved. Gone are the grainy, washed-out images. However, the front-facing camera hasn't been upgraded and is what I use the most for Skype and FaceTime. Perhaps Apple will upgrade the front-facing camera in the next release.
Update 19 March: One of the features I'd most like to see in an upcoming version of iOS is multiple user accounts. My new iPad is strictly for personal use and shared among 5 family members with different needs/applications. It's difficult to share an iPad today with multiple users and switch between different accounts (e.g. Twitter, FaceBook, DropBox etc.). Perhaps this is one reason Apple is selling so many iPads (one per person).
From left to right, the original iPad, second generation and the new iPad (white) with HD display