There may be a "trick" to this, depends on how those two headphone systems are connected. I suspect the cheaper gaming set is NOT plugged into a jack on your sound card. They may be plugged into a jack on the back panel of your mobo, or into a socket on the front of your case, or they may be part of some sound system that plugs into a USB port. You should tell us that detail.
The "trick" part is due to a limitation of Windows and you can deal with that easily. No matter how many different hardware devices your system has installed for sound, Windows can use only ONE of them at a time for sound playback. Similarly for sound recording, only one device can be used at any one time. Go into the configuration tools for Windows to where you specify sound settings. Look at the Sound Playback area, and you'll see that it is set (probably) currently to use your sound card for that purpose. In that case, all sound output will come from jacks on your sound card, and the gaming headphones (IF they are plugged in elsewhere) will be silent. But if you opt to CHANGE which hardware device is used to output sound - to, say, the mobo built-in system (often by Realtek) or a USB-connected device, then sound will come from that source and your sound card will not be used. You can come back here and change this setting any time you want, depending on what you want to listen to and how. A similar setting is available for what system is used for Sound Recording, and another also for MIDI playback if you ever do that.