The trial is fully functional, so purchasing won't change things.
Many people use HDHomerun tuners, so it will be something on your system. For example, the firewall, as Jim has explained. I know Windows 10 will turn the firewall back on if you don't have an active firewall. I'm not sure about Windows 8.1, but I suspect it does also.
Anyway, I'm not a HDHomerun user, and different models have different issues, so it would be a good idea to post the exact models that you have, so other people can assist.
In the mean time, conflict with another program is a common cause of issues with any tuner. You Windows 10 system works, and Windows 10 can't have Windows Media Center on it. What about your Windows 8.1 system, does it have WMC on it? If so, you need to stop all WMC tasks and services. You can't just close the program, but must stop its background services that do things like collect EPG data and start and stop TV recordings. In fact, it is best to uninstall WMC on a PC where you want to run [JRiver] MC. There is even a setting under "Tools>Options>Theatre View>Behaviour>Disable Windows Media Center" to stop WMC from stealing Infrared commands from MC.
If you have any other software accessing the tuners, even if it isn't active, you need to close it down.