The difference is that in traditionally in gaming you get "HDR Rendering" which is forcing bright lights to be darker and darker areas to be brighter in order to make everything fit within a preset brightness range of a standard display. This is done because the display chain and display itself was not designed to handle brightness levels beyond the confines of a traditional video signal. Think of it as compressing the dynamic range into a much, much smaller range.
True HDR gaming on HDR displays means that the brights stay bright and the dark stays dark. The HDR data is sent as a separate metadata stream within the video signal to be interpreted by the display. It's an extremely difficult thing to demonstrate without seeing it in person. If you record a video in HDR and play it back on a standard display, then everything will just look overly bright/washed out, but in reality, it's ****ing gorgeous. Of course, the effect will vary according to the abilities of your display.
Some LCD HDR TVs can get very bright (over 1,400 nits), but can't get very dark. OLED HDR TVs can't get as bright (800 nits), but have perfect blacks (0 nits). What looks better depends heavily upon the room the TV is used in.