The short answer
Line balancing is the objective: distributing work across stations so the line meets demand with minimal waiting and waste. Yamazumi is one of the instruments used to reach it: a stacked bar chart showing what work sits at each station, classified by type, measured against takt.
One is where you are going. The other is how you see where you currently are.
A Yamazumi chart answers "where is the imbalance, and what is it made of?" Line balancing answers "what do we change, and does the line then meet demand?" Neither question is useful without the other. Balancing without a chart is redistribution by intuition; a chart without balancing is a picture of a problem nobody fixed.