Deciding how to set up your iPhone is not as simple a choice as it may seem at first glance. If you have a backup from an iPhone or another iOS device, restoring your settings from it will save you the hassle of configuring everything again. Setting the iPhone up as a new device, though, does have its benefits and may even help you troubleshoot an issue.
Setting up your iPhone as a new one essentially means starting from scratch. At the end of the process, the iPhone will only have the default apps installed, and you will need to configure settings, set up email accounts and so on yourself. The only exception to this is if you already have an iCloud account to which you have synchronized some information, such as contacts, bookmarks or calendar events. In that case, that information will be copied to your new iPhone after you sign in to your account.
Resetting your iPhone to factory settings and then setting it up as a new one can be extremely helpful in narrowing down the cause of an issue when troubleshooting the phone. As long as you create a backup right before resetting the iPhone, you won't lose anything. If the issue persists after you set up your iPhone from scratch and without logging in to your iCloud account, faulty hardware is likely to be the cause. Otherwise, an app or your iCloud data are the source of the problem.
Restoring From Backup
If you have previously owned an iOS device and created a backup for it, either using iTunes or through iCloud, you also have the option of setting up your iPhone using that backup. A backup contains your custom information and settings, including contacts, bookmarks, calendar events and app data. It does not, however, include the apps themselves, nor your music, videos and other media -- you will have to sync those to your iPhone manually after the setup process is complete.
Having a backup doesn't necessarily mean you have to restore from it. If you use iCloud or have previously synced your old device with iTunes, you will be able to copy the important information, such as your contacts, calendar events, notes and reminders, to an iPhone that has been set up as a new one. Similarly, you will be able to redownload or sync apps and media you purchased through the iTunes store as long as you are signed in to the correct Apple ID. You will, however, lose application data and need to configure your iPhone again.