A heraldic badge is not a part of an armiger's coat of arms, but it is granted by the same body as a coat of arms.
The badge is designed to be used by the supporters of an armiger. A mediaeval knight would have a coat of arms, but his supporters or followers would not be allowed to use the coat of arms, instead they would use a badge to help identify themselves to each other and to other people.
Today, many English towns and cities have a coat of arms. Usually the mayor and council are the only ones allowed to used the full coat of arms. The departments of the council use a badge, and it is the badge which is painted on council vehicles etc. to identify them.
Badges, like coats of arms, can only be used with the permission of the person to whom they were granted.
The "Prince of Wales's Feathers" is the badge of the heir apparent of the British monarch. The badge is used because the coats of arms of the prince and the monarch are very similar.