If an app runs on the iPhone 4, will it run on the 4th generation iPod touch?
Solution 1:
As a developer, you can control exactly which hardware you want to support and which you will not support by picking the source code you use and properly tagging the files you upload to Apple.
The iOS app store then can prevent the devices you have disallowed from downloading the app. So, there's nothing to prevent a developer from targeting only iPhone 4 with their code or just targeting a specific iOS version.
It's up to you (and each other developer) to decide on a build-by-build basis, so the general answer would be:
No, there is no guarantee that all apps (or any one app) that run(s) on iPhone 4 will also run on iPod Touch (4th generation).
Solution 2:
You mention that they're having trouble downloading/installing the app — does the app require iOS 7? The iPod touch 4th generation can't run iOS 7, so apps that require iOS 7 won't install.
The main hardware limitation of the iPod touch over the iPhone (other than the lack of cellular connectivity) is GPS. All generations of iPod touch don't have 'real' GPS.
For iPod touch with Maps, the Maps application provides your approximate location using information based on your proximity to known Wi-Fi networks (when on and available). The more accurate the available information, the smaller the circle identifying your position on the map. The feature is not available in all areas. Known Wi-Fi networks are predominantly in urban areas.
Source: Apple — iPod touch — Technical Specifications
The iPod touch also doesn't have a compass.