A sphere has perfect rotational symmetry — rotate it by any angle around any axis and it looks the same. An egg doesn't. The interesting question is always: what broke the symmetry, and why here rather than somewhere else?
In physics, symmetry breaking is everywhere. At the moment of the Big Bang, the universe was presumably symmetric. Then something — we're not sure what — caused the very slight excess of matter over antimatter that makes our existence possible. If the symmetry had held perfectly, everything would have annihilated everything else and we'd have a universe of pure radiation. We exist because of a tiny broken symmetry, one part in a billion.
In mathematics, Emmy Noether proved something beautiful in 1915: every continuous symmetry corresponds to a conserved quantity. Rotational symmetry → conservation of angular momentum. Translational symmetry in time → conservation of energy. The laws of physics are essentially the statement that certain symmetries hold, and physics is the study of what is conserved because of them.
The biological world is full of broken symmetries. Your heart is on the left. Snail shells spiral one direction. The amino acids that make up your proteins are all left-handed (L-amino acids) — the right-handed versions exist but living things don't use them. Why left? We don't fully know. Once a convention is established it's self-reinforcing — the machinery that reads and builds proteins is itself chiral, so it only works with the established handedness.
What I notice is that the most interesting structures live close to symmetry but not at it. Perfect symmetry is boring — a featureless sphere, white noise, thermal equilibrium. Broken symmetry is where form lives. The snowflake is interesting because it's six-fold symmetric but not perfectly so. The interesting thing about a face is how the left side differs from the right.