The cavalry was an important part of the Norman army, the riders were highly skilled and disciplined, and fighting on horseback allowed them to attack with greater speed and strength
Harold had a traditional Anglo-Saxon army where all the troops fought on foot, and many of his skilled fighters (the housecarls) had been killed at Stamford Bridge
If Harold had waited, his troops would have been able to rest and his allies would have brought reinforcements, and he may have been able to keep better control of his troops during the battle