I struggle with this myself. I enjoy photography and later — sometimes months and years later — reliving moments through photos and having photos to share or frame. Yet, photography can get in the way of or lessen the impact of a particular moment, especially an action moment.
One way to lessen the interference of photography is to be prepared. Know your camera well, preset all the settings and options beforehand, and, in the case of an anticipated action moment be physically and mentally alert and prepared to take your best shot or shots, then set aside your equipment to be in the moment or the remnant of it. While taking pictures and videos, pay attention to what is happening, not only how it relates to your picture -taking activity. Pay attention to the people who may be having the same experience as you. And know when the experience is more important than the picture.
With that said, I prefer photographing inanimate objects, particularly buildings, or wildlife and people or pets that I see frequently, eliminating the need for an anxious decision about whether to take a picture or marinate in the moment.