They are different things.
A tire rotation means moving/swapping the front and rear tires to change their positions on the car, sometimes also swapping them side to side as well. Tire rotation logic can get complicated, but the basic idea is this: Your front tires do All the steering and 80% of the braking work, and all the acceleration work if it’s a front-wheel-drive car like the majority of cars. Since the rear tires do so little work compared to the front, they do not wear out nearly as fast, so you swap the front to the rear several times during the life of the set of tires in order to maximize tire life and keep the handling balanced during that tire life. How often the tires should be rotated is different from vehicle to vehicle, but is typically between 7,000 to 12,000 miles.
A balance is meant to fix shaking that is usually greatest around 60mph, but may come in and out at other speeds as well. Each wheel/tire assembly is put on a computerized machine that spins it and where around the edge of the wheel to attach weights so that the wheel is balanced as it spins. It’s normal to have to have wheels rebalanced at least once during the life of the tires, although sometimes tires will go all the way to the end of their life without wearing in a way that creates a shake that needs to be balanced out. If a wheel or tire is damaged by hitting something, a tire defect causes it’s shape to change, or the tire changes shape over times (wears unevenly) because of a alignment or shock absorber problem, balancing can make it more comfortable to continue getting the maximum lifespan out of the tires.