Chris,
I have experience working both with and without flash for bird photography.
As others have pointed out, sometimes photos that were taken with flash have an obvious quality that could be called distracting or artificial. But in my experience, that is not always true at all. Generally the trick is to adjust the power of the flash to balance the light so that the flash is only supporting the natural light, not overwhelming it. I find that a good starting point is to set the flash at -1 2/3 EV compensation. What flash adds to a photo is partly a subjective judgement and different people will have different sensitivities to it, with some people being total purists who cannot stand any trace or hint of artificial light in an image. Nobody likes a photo that seems immediately unnatural. I'm personally sensitive to over-saturation of colors. But I also appreciate the idea that to some extent, all photos are merely artistic interpretations of reality, so I'm not afraid to slightly enhance an image for more impact. With that in mind, in some situations I think the advantages of using flash far outweigh the disadvantages.
Using flash is not for everybody. I think probably a majority of bird photographers don't use flash because most do not use a tripod and using a flash is more cumbersome without a tripod. If you're using a tripod, it's easier to use a flash bracket to get the flash off the camera, which helps with red eye, etc. Once you have the flash available, there is nothing forcing you to use it. You can still turn it off, and I often do. And sometimes I just want to go lightweight and I will leave the flash behind, and sometimes the tripod too.
Nikonmike provided a link above to a very old web site on bird photography. Some parts of this site may need to be updated, but a lot of it is still completely valid. Specifically,
I agree with the 3 benefits of using flash presented on this page. After not using flash for the first few years of my bird photography, once I tried using flash I was immediately surprised that it sometimes dramatically improved both the
color and yes, the
detail in my images. So for me, for some images, adding flash to my system has "raised the bar" for what I consider to be a really good bird photo.
Good luck to you!
Dave