For what you are trying to do, either would work (I use both too).
Flickr and other photo sharing sites are not really designed/intended for temporary storage where you delete photos after a while but certainly you could do that. With Dropbox you can create a group shared folder and park all the files there and your designated family members can sync directly to their computers. With Flickr they would order prints or download the files from their web browser - I don't know of a way to quickly download hundreds of files all at once on Flickr though - so Dropbox may be your best choice.
Both are free up to a certain size limit which I forget, but it's pretty huge. I have paid pro accounts on both since I use them so heavily. I'm
cjcam on Flickr.