Here's some tips we've been able to figure out from talking with users:
First, make sure you are logged into the Apple App Store account as you were under your old iPad. Then, using the App Store app, check your purchase history. If Photosmith isn't listed, please read on. If it is listed, download the app from there.
To solve this issue, we suggest you contact Apple directly for assistance via: https://expresslane.apple.com/Issues.action
We wish we could help you directly, but we don't have the access necessary to troubleshoot - Purchases made through Apple's App Store are (unfortunately) between you and Apple - we don't know who our users are until they contact us for help. Thus, we don't have any ability to validate your purchase or fix your account with Apple.