Asymmetry shows the differences and similarities of two objects. This helps create a nicely laid out photo that shows the comparisons and the staging of the two objects.
Balance in photography is when there are two objects that have the same visual weight but are different. They can be the same object with different details, such as the photos below. Photographers use the balance to show some symmetry but have the details be different so that it is more eye pleasing.
Symmetry is when two objects are the same. When you draw a line down the middle, they resemble the same objects with the same framing and detail.