Flickr is a bit more commercial grade, but if you want to share photos with family and friends (or students) this is a great tool.
Picasaweb is part of the Google suite and not to be confused with Picasa the image editor. That is a COMPLETELY different thing. Picasaweb shares images online. Picasa the free software edits photos. Here are my photos that I have elected to share publicly on Picasaweb.
Flickr allows you to embed content nicely and so does Picasaweb. Flickr is owned by Yahoo, and that could spell trouble. But Google and world domination is always worth wondering about too.