These errors all mean the same thing: iTunes can't connect to Apple's servers. The restore or update process requires server verification, and something is blocking that connection.
Good news — these are almost never hardware problems. They're network/software issues that you can fix yourself.
The errors and what triggers them
Error 17 / 1638: General Apple server connection failure. Firewall, DNS, or internet outage.
Error 3000: No internet connection detected at all.
Error 3004: Firewall or proxy is specifically blocking Apple's update servers (gs.apple.com, albert.apple.com).
Error 3014: Apple's server is overloaded — common right after a new iOS release when millions try to update simultaneously.
Error 3200: The firmware file (IPSW) didn't download completely due to a network interruption.
How to fix all of them
Check your internet connection
Open a browser and try loading apple.com. If it doesn't load, fix your internet first. Restart your router/modem. Try a wired Ethernet connection instead of Wi-Fi.
Disable firewall and antivirus temporarily
Security software often blocks iTunes from connecting to Apple. Disable it, try the restore, then re-enable it afterward.
Change your DNS to Google (8.8.8.8)
Your ISP's DNS server might be blocking or failing to resolve Apple's servers. Switching to Google DNS (8.8.8.8 / 8.8.4.4) or Cloudflare (1.1.1.1) can bypass this.
Check your hosts file
If you've ever used jailbreak tools, they may have redirected Apple's server in your hosts file. Open it and delete any line containing gs.apple.com.
For Error 3014 — just wait
If a new iOS was just released, Apple's servers get slammed. Wait a few hours and try again.
For Error 3200 — delete IPSW and re-download
The firmware file is incomplete. Delete it and let iTunes download a fresh copy. Make sure your connection is stable during the download.
If you've tried everything above and the error keeps coming back, the problem is most likely physical — something broken inside the device that software can't fix. That's when it needs professional hands-on inspection.