This distinction is more than just semantics. There are indeed two different uses for your camera when you go birding. Some use one or the other; I use both. When I was new birder I marveled at the skill shown by others in making field ID’s, especially for those small, constantly moving birds, often in the high canopy. I found that, especially when birding alone, taking a picture and deferring the ID until home, when I had the time and ability to zoom and enhance the picture, was very helpful. These are not great pictures and not kept in my picture library, but merely utilitarian shots to help me learn the field marks and expand my skills. Sometimes the picture was just inadequate for the task–oh well. This is photo-birding. The down side to this technique is if you let the process of taking the picture distract you from observing the bird and learning to find field marks quickly. Sometimes you’ll find that the bird has flown while you’re still reaching for the camera. Purists may scoff at this technique, but it has worked for me.
Bird photography, on the other hand is the process of making the perfect exposure for a bird–something that you would be proud to post or publish. It may be a common bird, but you’re striving to catch it feeding, mating, flying, etc. with the perfect composition and exposure. The possibilities are endless. Adding this to my birding adventures has created some additional interest for me in this wonderful hobby. Even when there are no rarities at hand, you can work for that perfect picture.