In both cases, the reason for authorizing the use of slaves in the armies was because they thought that using the slaves would help them to win the war. Both sides felt that they needed the extra manpower to try to win the war.
In both cases, the slaves who fought were often offered the chance to become free. They were told that they would be set free if they would fight in the war. In some states, slaves were actually bought by the government and freed to go and fight in the war. This was because the states all had quotas of soldiers that they needed to provide and could not always fill them with white men.
Overall, then, both the British and the Americans used slaves in the war to provide extra manpower that would, they hoped, allow them to win.