I am moving my photos over to Flickr from Picasa Web Albums and gradually changing all of my links - I thought I would share my personal opinion on the two services as a very casual user:
Flickr Good - More people and better community!!! Bad - There is not a full featured desktop client for Flickr (that I know of...) like Picasa - for me organizing and tagging is slightly painful in the Flickr web interface (compared to using Picasa on the desktop). [Lance points out below that Picasa is still useful for tags and captions if you do those before you upload!] If sharing and community features are important to you I think Flickr is the better service.
Picasa Web Albums Good - Decent integration with Picasa makes it easy to organize/tag/caption pictures on your desktop and then transfer them to Picasa. Bad - Picasa does not have the same level of community or number of members that Flickr has. If your primary concern is online personal photo storage and occasionally sharing photos then the integration with Picasa makes this a strong choice.
Other ideas? CM