Alas, in the years since this article was originally published, Picasa has been deemphasized in favor of Google+, and Flickr has lagged behind in development. Both have changed their pricing. Today, neither is a clear winner.
I like Picasa, and I like Flickr, and if you'd asked me a couple years ago who has the better photo storage site, I'd have gone with Flickr hands down. These days, I lean toward Picasa. Let's break it down and explore why Picasa may be the better place to put your pics.
The Many Confusing Photos of Google
It's a bit confusing, but Google actually has several items that can handle photos. There's Picasa, a desktop photo editing app, and Picasa Web Albums, the online extension of Picasa. Most people just refer to both items as Picasa. You don't need to use the desktop version in order to use the Web version, and vice versa. Picasa Web albums is also the hidden photo storage back end for Blogger and Google+, so you may be a Picasa user without realizing it. Google also owns Panaramio, which is a photo sharing site organized around geographic location. To add one more layer of confusion, Google also has an online photo editing app called Picnik, and its technology powers the photo editing features in Google+. You can also use Picnik to edit photos in Flickr.
Clear as mud? When I say Picasa, I'll use it mainly to refer to the online photo sharing site unless I specify I'm talking about the desktop app.
One of the reasons Picasa used to be the inferior product is because it lacked features Flickr has had for years. Now both products share many common features. Both apps allow you to specify different levels of privacy for your pictures. Both apps allow you to tag people in your photos. Both allow labels, albums, printing, and geotagging (associating a geographic location with a photo, which is often done automatically by phone cameras and other devices).
Both offer free and paid account options. You can print photos or order online prints from either app, and you can bulk upload your photos, embed them in other websites, or comment on photos others have posted. You can even generate slide shows. You can specify Creative Commons licenses or retain all copyright protections for your works with easy settings that you can change on a site-wide or per photo basis. Both allow you to track the number of views and overall popularity of your photo, and both allow video uploads.
In other words, both Picasa and Flickr offer the basic features that most people would want in an online photo album.
Flickr is an established player. It's been around for longer, and it's still used by a lot of serious photographers. There's no point moving to a different service if you'd have to leave all your friends to get there. Flickr also offers photo sharing groups, which is something Picasa doesn't directly provide.
Picasa is tied into Android phones and most other Google products, so it's integrated into a lot of the services you may use anyway. It's also easier to move your photos from Picasa into another app. You can send photos to Picnik with a single click (a feature that was disabled in Flickr when Google bought Picnik), and you can download entire albums all at once.
Because Picasa is also tied to a desktop app, you can use it to manage the photos on your hard drive and automatically sync them online. Phone camera photographers can also rejoice, because smaller photos don't count toward your file upload price.
When you press the "order prints" link, you're not stuck with only one printing service. Picasa lets you chose from many popular photo printers, including Snapfish, Shutterfly, and Walgreens.
Picasa and Flickr have two different pricing schemes, so your needs will dictate which is the better deal. Flickr limits free accounts to 300 mb uploads per month, with a 15 mb limit per photo. You're also limited to 10 group pools and can't access your original photos to download them later, and you can't see past the first 200 photos in your photo stream. There are other restrictions, but those are the basics. A pro account is $24.95 a year and unlocks unlimited uploads, infinite photos in the photo stream, and up to 60 groups for sharing. Pro accounts also remove the ads from Flickr and give users view count statistics for their photos.
Picasa operates mainly on a storage fee. Small picture uploads don't count towards your quota, so you can upload as many 2048 x 2048 (around 4 megapixels) photos as you'd like, and photos uploaded through Google+ are automatically resized to fit within that constraint. Photos over that size count towards your currently 1 GB quota, but Google will resize photos after you've reached that quota. If you're interested in ordering prints or storing archival quality prints, you can order more storage space at prices starting at $5 per year for 20 GB of extra space.
If you're a prolific photographer with lots of large images, the unlimited plan at Flickr is probably the better deal, but users who primarily just want a way to display their photos online may find that Google offers the better bargain.
The Final Assessment
Both Picasa and Flickr are solid photo sharing sites with great features. However, Picasa has been growing in leaps and bounds with new features. Flickr is pretty close to the same service now as it has been for the last several years. Unless they find a way to innovate, Picasa is going to offer a clearly superior product within a few more upgrades. I'm still a Flickr Pro user, but like Flickr itself, that may be more due to inertia than it is the current feature set.