Facebook App Type: Which one to pick for mobile AND web?
TL;DR: use 'web' unless you're putting the app secret into the iOS / Android binary
The primary difference (and also the reason that it says 'it doesn't matter' while also giving a recommendation) is that if you choose 'native/desktop' mode, it's assumed that you distribute the app binary with the application secret embedded and the app secret is considered untrusted.
In this configuration, API calls which use the app secret (e.g. creating test users, making API calls on behalf of the app itself like changing app settings, posting achievements, etc) will not work.
If your app doesn't need to use the app secret / app access token, setting it to native/desktop increases the security somewhat as there's one less attack vector for your app.
If you're distributing the app secret with your code you MUST use that option or users could easily determine the app secret and hijack the app itself.