In your original post you stated that you wanted to photograph chamois and deer. If you are mostly doing handheld shooting, then consider the size and weight of the AF 300/2.8 lenses. Technically it is better than the 300/4 lenses but for practical purposes it may be too big and heavy. It may be fine for short periods, the extra stop in speed enables you to use faster shutter speed for models without VR. But it must be heavy for extended times, these lenses are about double the weight of your AF 300/4. A smaller lens like your 300/4 or a lens with VR may be much more useful.
For hand-held shooting the AF-S 300/4 is probably about the same optically as your AF 300/4. You would need very good technique to show the improved optical qualities of the newer lens. Some have reported the bokeh of the AF-S 300/4 is not as smooth as some other models. The AF-S 300/4 is quieter so it is less likely to scare wildlife (if the camera shutter hasn't already scared them
) The AF speed on this lens is not particularly fast, the 300/2.8 versions are much faster. The AF-S 300/4 is marginally lighter than your lens with the tripod foot removed, but heavier with the tripod foot. For handheld shooting some photographers have reported the 70-300VR zooms give them more keepers than the AF-S 300/4, VR more than makes up for the slower aperture. If the aperture and reach are enough then maybe the AF-P 70-300VR is a better option, maybe consider a D7200, D7500 or D500 if you need more reach?
On the other hand, if you can shoot from a tripod or monopod for much of the time, a 300/2.8 could be a better option. The extra stop in speed will allow you to photograph in deep shade or twilight conditions more easily, and will give you more subject isolation (provided the depth of field is sufficient for the subject).