Using Jihosoft Phone Transfer
As a matter of fact, with a phone transfer software, you really do not need to worry about transferring data, like contacts, videos, music and more iPhone to Android.
As for phone transfer software, I recommend Jihosoft Phone Transfer. Jihosoft Phone Transfer can help you transfer photos or videos from iPhone to Android without any internet connection, and you can complete the transfer in one-click.

1.Install Jihosoft Phone Transfer on your PC, and run it.
2.Connect both of your iPhone and Galaxy S4 to your PC via USB cables.
3.Jihosoft Phone Transfer automatically set Galaxy as “Destination”, and iPhone as “Source”.
4. Choose "contacts”, and click”Start Copy” to transfer. So easy

This is quite complete. Thanks!

There are multiple methods of transferring contacts from an iPhone to the Galaxy S4 Mini. Below, you will find four of the simplest guides on how to transfer contacts. Choose whichever method you find more suitable.

Using iTunes to Transfer Contacts from an iPhone:

1. On your computer, launch iTunes
2. Connect your iPhone to your computer using the USB cable
3. In iTunes, click on your iPhone once it appears
4. In the device's summary page, click the "Info" tab
5. Mark the "Sync Contacts With" check box and choose "Google Contacts" from the drop-down menu
6. Enter your Google account and password if this is the first time doing these steps. If not, then click "Configure" and select a previously added Google account
7. Click "Apply" and let iTunes sync the contacts from your iPhone to your Google account
8. On your Galaxy S4 Mini, sign in with the aforementioned Google account if you haven't done so already. If you need to add the contact, go into "Settings" and tap "Accounts"
9. Allow the Google account to sync the contacts on your SGS4 Mini

Using iCloud to Transfer Contacts from an iPhone:

1. Use your computer's Internet browser to access www.icloud.com
2. Sign in to your iCloud account
3. Click "Contacts" to access the list of contacts
4. Select the contact(s) that you wish to transfer / export
5. Click the button depicting a gear (located in the lower-left side of the window) and click "Export vCard"
6. After your computer downloads the VCF file, import it to Google Contacts
7. Alternatively, connect your SGS4 to your computer using the USB cable and copy the VCF file to your phone's "/sdcard" folder. Then, access the "Contacts" application on your smartphone, press the "Menu" button and tap "Import / Export" > "Import from SD card"

Using the "My Contacts Backup" to Transfer iPhone Contacts:

1. Install the "My Contacts Backup" application on your iPhone. You can find it on the App Store
2. Launch the application and tap "Backup"
3. After your contacts will be backed up, tap "Email"
4. Use your own Gmail address as the recipient for the email containing the attached VCF file
5. Send the email, and open it on your SGS4 Mini
6. Download the VCF file from the email onto your Mini to import the contacts

Using the "Bump" Application to Transfer iPhone Contacts:

NOTE: an active Internet connection is required on both smartphones in order to use Bump for transferring contacts.

1. Using your iPhone, go to the App Store and download the "Bump" application
2. On your Galaxy S4 Mini, access the Google Play Store and download and install the "Bump" app
3. Launch Bump on both smartphones and allow permissions if / when prompted to do so
4. On your iPhone, access the "Contacts" section in Bump
5. In the list of contacts, select the ones that you want to send to the SGS4 Mini
6. When the label "Bump now!" appears in the upper-right corner, you can lightly bump the two handsets
7. Tap "Connect" and let the devices finish the transfer process
8. A copy of the previously selected contacts will be transferred from the cloud to the SGS4 Mini