It's hard to state what the issue is here without going into some SE.SuperUser territory, but I believe one of these suggestions will fix your issue. You've tested and determined that the server is reachable from a different PC on the same network. It's clear that the problem isn't with Minecraft as it's only affecting first PC and only when connecting to a third-party server. We'll need to figure out if it's a problem with Windows or a problem with your home router.
- Turn off Windows Firewall
Assuming you're using Windows 7+, you'll want to see if turning the built-in firewall off fixes the issue.
To disable the firewall portion of Windows Firewall with Advanced Security by using the Windows Firewall Control Panel program
Click Start, click Control Panel, click Network and Internet, and then under Windows Firewall, click Turn Windows Firewall on or off.
On the General tab of the Windows Firewall Settings dialog box, select Off (not recommended), and then click OK.
If this fixes the problem, you'll want to re-enable the firewall, and setup a firewall exception in Windows.
- Place the PC on the router's DMZ
Assuming that turning the firewall off didn't work, this should be your next step. In this one you'll have to research how do it as it's different steps and locations even for the same brand and manufacturer. Generally speaking though, a home router will offer the option to place a device in DMZ, or "non-protected" status. The idea behind this is that all traffic will be directly to the IP address in DMZ first unless stated as a forward otherwise. This should only be for testing purposes. If it works on DMZ, then you'll want to check your forwarding settings again and see if the ports and destination IP is correct. Leaving a computer on the DMZ leaves it open to any and all connections, which makes it a possible target to people and bots with malice intent, but also removes any and all possible restrictions
- Repair Windows TCP/IP stack
If the other two options do not work, there's something most likely messing with outbound connections. At this point, I would see if repairing your TCP/IP settings would fix the issue.
Click on the Start button, type cmd and right-click on Command Prompt; slect "Run as Administrator" from the context menu.
Inside the new Command Prompt window, type:
netsh int ip reset
When complete, reboot the PC.
Repairing the TCP/IP stack should disconnect any additional drivers and leave only the windows one behind. If this doesn't fix the issue, there might be a bigger problem you'll want a repair shop to look into.