Exporting From Facebook to Flickr
There is currently no direct way to export your photos from Facebook to Flickr. An app has been created to import from Flickr to Facebook, but the reverse does not exist yet. However, there are ways to create a work flow that would make uploading to both Facebook and Flickr easy and simple if you have the right software.
If you have a Mac and use iPhoto to upload your images to Facebook, then you can easily upload these images to Flickr. You follow the same steps as you would for a Facebook upload; the only difference is that you click the Flickr button instead of the Facebook one. Enter your Flickr account information and then your upload begins.
If you don't have iPhoto, the process can be more complicated.
Facebook compresses your images, so you might what to upload your best photos from the original photos to preserve quality. Although this is time-consuming, it might be worth it if your goal is to showcase your work as a photographer.
If the original file of your image is lost, then you can download it off of Facebook by right clicking. Once the image is on your hard drive, you can upload it to Flickr.
An app doesn't exist yet to make the process quick and easy, but until then you can still get by.