Engine mounts typically let the engine rock back and forth too much and can lead to clunking noises. They can also let one end of the engine sit too low. In severe cases one of the pulleys or the belt can rub on plastic splash shields. Sitting at the wrong height also changes the angle of the half shafts. On high-mileage cars that can lead to a shimmy in the steering wheel from worn areas inside the inner cv joints. In real bad cases I've even heard of inner cv joints falling apart causing you to sit on the side of the road in a puddle of tears.
It all depends on which mount is broken or worn. Typically one on each side holds the engine up and one or two in front or back keeps it from rocking but that's all. If it rocks too much, besides clunking, it can open up a gap in the spring-loaded exhaust gasket. You'll hear that as a loud clicking noise during periods of acceleration.