2013 Mac Pro network connection issues

It may be possible that the network configuration settings on this device has become corrupted.

I suggest trying the following:

1. Create a New User Account
   1. System Preferences > Accounts
   2. Click "+" to add a new account.
   3. Log off, and then, log back on with the new user account.
   4. Try connecting to Internet.
   5. If this resolves the problem, there may be something amiss with the current user profile and will require further troubleshooting.
   6. If this does not resolve the problem, go ahead and log back into the "normal" account and go on to the next step.
2. Move System Configuration Files
   1. Open Finder > Go > Go To Folder
      1. In the "Go to the folder" window, enter:/Library/Preferences/SystemConfiguration/
      2. Click on Go
   2. Locate and select the following files:Note: Files are macOS version specific, not all the following files are used in each version:
      1. com.apple.network.eapolclient.configuration.plist
      2. NetworkInterfaces.plist
      3. preferences.plist
   3. Move any matching files to the macOS Desktop. You can place them all in a folder as a backup. The "missing" files will be rebuilt after restarting the Mac.
   4. Reboot the Mac.
   5. Try accessing the Internet again.

The fact that your Mac Pro can successfully access the local LAN, but not the Internet, verifies that there is nothing wrong with either your Mac's Ethernet or Wi-Fi interfaces.

Loss of Internet access, could be for a number of reasons, including:

1. The Mac network interface is not properly configured. Is it currently configured as a DHCP client (the default) or has been reconfigured to use static IP addressing?
2. Your company requires the use of a proxy server to allow local network devices access to the Internet. The settings for this may be missing or misconfigured. You will need to talk with your company's IT folks about this.
3. Your company may be blocking Internet access via their router or firewall appliance. Again, you would need to discuss this with IT.
4. The company router/firewall may be misconfigured or faulty.
5. There is an issue between the company router/firewall and the ISP.

If you enter:

17.142.160.59

... in the search bar in your Web Browser, and you get to Apple.com

then enter:

apple.com

... in the search bar and you get nowhere...

you have a Domain Name Server (DNS) problem.

You're very welcome ... and thank you for posting your solution. I'm sure it will help others with the same issue.