Tips for Using the Clone Tool to Remove Flyaways
Cloning generally works better for flyaway hairs. In fact, when I am trying to remove something from the background, cloning is generally the option to choose anytime there is a lot of contrast between the background and the object you are removing.
Tips for Using the Heal Tool
Healing works best when you are trying to blend something and don't want/need an exact match. Healing will blend the area you are fixing with the area Lightroom chooses (or you choose by moving the spot Lightroom chose).
Healing also works best when you are working with an area that is very similar in tone to the area you are fixing. Blemishes are a great example of this.
Tips for Using Both Tools
Play with the size and feather of your cloning/healing brush. The closer you are to a hard edge the less feather you will want in your brush.
If you are hoping to remove something large, it generally will be best to work in smaller chunks and not with just one large clone/heal job. This also works best if you have a limited "good" area you are pulling from.
Special thanks to Amy for creating these super helpful videos showing how she uses both these tools to remove distractions from an image.