^^^ What he said. Excellent explanation, Overread.
I'd add that if the scene you're shooting pretty much averages to 18% grey, you can go with the meter. If, it is at least 1 stop darker or lighter, then you'll be fiddling with the exposure compensation if you're shooting Aperture priority. If the scene is consistent in its tonality and lighting, then you can dial it in, check the histogram, and you're good to go. If you're facing rather large differences (say, sun-dappled forest floor, or the sun playing peek-a-boo behind clouds), you may have to go to spot meter mode to be sure you're getting the right reading.
#2 in Overread's list is very important. When you're shooting a series that will be looked at together, it can be disconcerting to see large differences in exposure within the same series, and in this situation, it may be better to have a single exposure for the whole set. For instance, one of my uses of photography is to document external house inspections and wherever possible, I use a single exposure so that as we move from image to image in viewing with the clients, we see consistent tones in the same objects. In setting this single exposure, I do take an incident meter reading, as well as checking specific spots with a spot-meter. This gives me an exposure value where everything I want to be able to see will have some detail in it.
In deciding between the two modes, I use manual in situations where I have the time to get the exposure right, and when I need consistency of exposure. I use Aperture-priority when things are fast-moving, or light is changing , or the scene tonality varies all over the place, and getting "a" shot is more important than getting the "perfect" exposure.