Since both Picasa and Flickr are free in their standard versions, you don't have to choose. I use both. The two are different enough that I find the strengths and weaknesses complement each other. A long time ago I opened a Flickr account and let it languish unused. Google's Picasa came along and based on the Google brand, I signed up and started using Picasa as my main photo management tool.
With the Picasa tool, you download the software and install it to your computer. After a few journeys into Picasa I quickly learned to appreciate its ability to help me keep track of photos. There have been several upgrades since I first jumped in and each upgrade has added incremental improvements. Face recognitions is cool. But being able to quickly find a photo on the hard drive is a wonderful feature for me.
Picasa's photo editor is also a time-saver and offers a full set of commonly needed photo editing tools such as crops, highlights, sharpen and even the cute "I'm feeling lucky" exposure adjustment works in a surprising number of instances. For the heavy lifting or to indulge in deep image manipulations there's still Photoshop.
The online "Picasa web albums" feature is a great way to back up your photos and to share them with other people. From a business perspective, I use Picasa online to deliver photos to my freelance clients. There's an upload option that allows you to choose if you want to upload full resolution images or other less bandwidth intensive versions. You can choose to make the albums public, private or hidden. For only a few dollars a year you can upgrade your account and get something on the order of 80 GB of storage.
Flickr and Picasa part ways the most online. In Flickr, the photo editor is online. The Flickr editor works well but is a touch slower to use since it's online not on your computer. The edits you do are applied to the photo online so if you want a copy of your edited image you need to then download the file. All good, but I prefer the Picasa software for editing.
Flickr is better at showing photos online and far better in terms of "community." Flickr is a social networking program built around photos. In that regard Flickr is a lot more sensible than Facebook. But I digress. If you like to participate in the wide world of photography, Flickr is hands down better than Picasa. That's not a criticism of Picasa because there are some social networking features available and they do work. It's just that Flickr beat Picasa to the punch and offers a greater sense of community and participation.
When it comes to sharing photos, the two programs are both excellent with only subtle differences. You blog photos from Flickr online and you blog photos from Picasa with the desktop software. In both it's a simple push-button process with similar options for sizing the photo. Both work well for photo hosting images for use on web sites. I do think I like the Flickr photo to blog options slightly better. The photo is hosted in Flickr and it shows up on your blog with a thin, black border. It's attractive. Picasa uploads the image to the blog for hosting. Not surprising, I guess, Picasa works most seamlessly with Blogger, another Google product.
A Flickr Pro account is $25/year and that gives you unlimited uploads and storage and a handful of features to help with marketing and having fun. For the storage upgrade in Picasa you pay $20 for 80 GB of storage and it's good for a couple of years. Google has continued to offer more and more storage space over time. My volume is pretty low compared to many people so by the time I get to 80 GB, Picasa will probably have unlimited storage, too.
You may use one or the other if you choose. Me? I think I'll keep using both.