As far as I know, Pear Note is working properly syncing documents on both iPad and iPhone. I have had several emails similar to yours, but all have been addressed by me explaining how Pear Note syncs and some of the limitations of its syncing.
On the Mac, Dropbox is always running in the background syncing things. On iPad/iPhone, Pear Note has to sync with Dropbox itself. It'll do this while you're running Pear Note and it has an Internet connection. It'll also try to keep syncing in the background for a few minutes after you leave Pear Note, assuming you still have an Internet connection. But iOS will only let it keep syncing in the background for a few minutes (10 or less). If it can't finish syncing, it won't try again until you launch Pear Note again. So, if you finish taking notes, throw your device in your bag, and walk out of Wifi range, your notes likely won't show up on Dropbox until you next relaunch Pear Note.
Internally, Pear Note syncs entire folders at once. So, if Pear Note for iPhone is trying to download from your Dropbox, nothing will show up until it downloads an entire folder. So, it's possible for Pear Note to get caught in an endless cycle where it never has enough time to download the first note or first folder before getting cut off from the Internet.
Note that Pear Note documents contain recordings which make them inherently large files. Depending on your Internet connection speed and how fast Dropbox is going today (which varies significantly), it can take some time to download. You can look in your Dropbox folder to see how much space your Pear Notes are taking up. I've seen people hundreds of notes taking up several gigabytes, which caused their initial download to a new device to take many hours.
If that doesn't help, please send an email to support and I'll walk you through tracking down the problem. If it turns out you have hit a bug, then I'll of course fix it in an upcoming update.