How to allow remote desktop from another subnet

On the Firewall create a rule to allow RDP or port 3389 in.  Also, make sure that Remote connections is enabled on the computer you're trying to remote into

Remote Desktop from another computer on the same subnet works perfectly.

Same as a ping, that works locally, but not remotely.

Right, however in Windows firewall you have three networks:

Domain networks
Private networks
Guest or public networks

If it works from the same subnet, thats because in the Domain network, you have RDP allowed.

However, the remote computer is not in the same domain, its either going to be considered in the private or public network.  

You'll need to create a rule for those networks to allow RDP.  You can specific a source and give the remote computer a static IP address and use that as the source.

In addition to the above suggestions, on the Windows 10 computer (Windows Pro I trust), make sure in the Network Sharing settings that Network Discovery is enabled.

I RDP to my client Windows 10 Computer and it works fine.

The one thing that hasn't been mentioned and MUST be set is the scope of the Firewall rule that allows RDP in the first place.
I see them set for Remote Address as ANY - so in that case it won't matter.

But, if it's set at LocalSubnet, then you need to add the other subnet - in addition to Local Subnet on the receiving computer at the least.

Same for PING, file and printer sharing, etc.  So, if ICMP is required as well (I'm a bit unsure about this) then add the subnet to its scope:
File and Printer Sharing (Echo Request - ICMPv4-In)
File and Printer Sharing (Echo Request - ICMPv6-In)
It can't hurt even if redundant...

Windows 10 Default Firewall rules have allowed RDP Connections into them. I have tested this on several machines. If someone or some application changed firewall rules that would be a different matter, but default firewall settings do work.

John:  Yes but PING still won't work - which was part of the concern.