I'm late to the party, but here's why I need (well, want) more than 16GB:
Apps - some are large enough and used frequently enough that I don't want to have to do the download/delete cycle on them.
Photos - I went to Italy a few weeks ago with a group. One of the guys had a 16GB phone. He was emailing the photos to his wife each night and deleting them after because he didn't have enough space. That's annoying, and what do you do when you don't have a data connection and run out of space?
Music - sort of agree, but again, limited data connections do happen. Even when they don't, streaming saps the battery compared to playing already downloaded songs.
Books - I play tabletop RPGs, less now than I used to. But when I played more, being able to pull from my collection of books quickly (some of these are just scans from old books, 100MB for a manual). Having to download them off Dropbox each time, or download/delete cycle through them to make sure I had enough space, would be annoying.
Updates - an ex-girlfriend had the 16GB one when iOS 8 (?) came out. I kept telling her that we could update it on her laptop via iTunes and get around the problem, instead she cleared out a bunch of photos (and later regretted it because she was a dope and didn't back them up first) and apps (fewer regrets, those were just downloaded again) so she could download and install the OS update. AS I recall, that one had an absurdly large free space requirement, but even requiring a more modest 500MB-1GB of free space could be difficult to meet sometimes if you only have 16GB and downloaded music, apps, photos, videos. Which items do you delete to make room?
EDIT: I'll note, this is more of an issue for me on tablets than my phone. I have fewer big apps on the phone that I couldn't delete, and I never watch video other than streaming some youtube content if someone sends me a link. My tablet, on the other hand, is my go to device when traveling and gaming (a fair number of good board games have apps for iOS and the iPad, much easier than carrying around several boxes of boards and pieces and rules to play with friends).