The best solution, IMO, is to use the browser as a platform: a Chrome extension can allow users to seamlessly sync data between services, because Chrome already has access both to your Facebook account and your Picasa account.
(I work on the Facebook platform.)
BUT - what if I actually want to export all of my photos into SomeApp.com, and I want to give SomeApp the right to store my photos indefinitely? Is there an API they can use to pull it from Facebook directly?
You can definitely export all your photos into SomeApp.com using the graph API and they can store your photos indefinitely. These APIs are documented at http://developers.facebook.com/docs/api.
When Facebook switched from their old API to their new OAuth-based API they also removed restrictions on storing data.
What Google is insisting on is "download my friends' data." This is where Facebook currently draws the line; we make it easy to retrieve your own data programmatically, while retrieving your friends' data programmatically is harder.
And, whatever Google says, Facebook is right to draw the line somewhere, and probably at friends' contact info. Consider an immediate consequence of doing what Google is insisting on: Zynga (e.g.) will spam all of their users' friends' emails. Because now they can retrieve them in bulk via a Facebook API. If you use Facebook, and tomorrow you started getting reams of email from applications you do not use, would you think, "How wonderful that Facebook has finally liberated my data!"? Or would you, maybe, be annoyed, and wonder why Facebook would ever consider making such a boneheaded decision?
[1] https://register.facebook.com/editaccount.php -> "Download your Information"
The user could sell out their friends to a third party if they chose to (opt in). But Zynga would only get access if the user said so.
The reason Facebook won't do it is that it would commoditize the social graph and allow straight head to head competition on features. I think no one, even Facebook, knows how that would turn out...
Except for Google, who clearly decided to make it happen.
(I work on the Facebook platform.)