Google details the new features in Picasa 3.9 in a Google Photos Blog post and a separate New features in Picasa 3.9 page on the Google Support web site. Basically it breaks down like this:
Google+ integration. Picasa now integrates with Google's Facebook alternative, allowing you to share directly to your Google+ circles. (Previously, the only sharing functionality in Picasa was with Picasaweb, which works well for online photo backup but isn't really the greatest online photo sharing solution.) Additionally, you can now upload and share name tags on Google+.
New photo editing effects. Google has added 24 new photo editing effects, including cross process, duo-tone, heat map, and many others.
Side-by-side photo editing. You can now compare two different versions of the same photo side-by-side so you can see how changes effect the original. (Alternatively, you can edit two different photos side by side as well.)
Free photo uploading ... at a sacrifice in quality. If you don't mind using Picasa's "Best for web sharing" option, you can upload any number of pictures to Google+ for free. I happen to prefer using Picasaweb for full-resolution photo backup, so I pay Google for extra storage. But if your primary goal is to share photos with people, as you might via Facebook or Twitter, this is an interesting option.
Aside from my Picasaweb uploading I probably use Windows Live Photo Gallery more often than Picasa, but since they're both free, and both are pretty excellent there's no reason not to have both installed. You can find Google Picasa 3.9 on the newly redesigned Picasa web site.