Net's down? Web app: Welp, you're SOL, son - Native app: Net? What net. Winner, native app.
Web applications can include an offline version using application manifests, IndexedDB, and an IndexedDB-compatible shim around the SQLite included in WebKit. Native applications can display an alert box and close if they fail to connect to the Internet service to which they were designed to connect. For example, good luck using the Facebook or Twitter application offline. Advantage neither.
Your data leaking due to hackers or court subpoena? Web app: Just cross your fingers and pray it doesn't happen. You probably won't know until too late, anyways - Native app: At least your data and its protection is in your own hands.
The court can just subpoena the server used by the native application's remote backup feature.
Whut, you thought setting up payment system for your site is that easy, requires no expenses to maintain and has no transaction fees?
Stripe, Authorize.Net, PayPal, and Dwolla charge transaction fees. Apple, Google, Microsoft, OUYA, and the like charge bigger transaction fees.