New iPhone 5s restore issue?

I just received a new iPhone 5s in the mail -- 32 gbs -- and I went to restore it from my old 5c. It worked well enough, but the new iPhone is already saying that there's not enough storage, which makes 0 sense to me. It is listing the total available space as 13 gbs -- not anything close to 32. How do I make the phone recognize that it has 32 gbs of storage (or something close to that)?

Connect your device to your computer using the cable that came with it.
Select your iPhone, iPad, or iPod when it appears in iTunes.
In the Summary panel, click Restore.
Click Restore again to confirm that you want to restore your device to factory settings and delete all data and content.