Thumb support usually refers to a type of smaller brace that only covers the thumb or finger. However, the medical industry seems to refer to thumb support and thumb brace interchangeably.
Thumb support is used to fix the position and support only the thumb or one finger. It is applied as a treatment for injury at a small part of the thumb or finger. Therefore, the thumb support only covers the affected area, leaving other fingers and the rest of the palm free to movement.
Nevertheless, as the term is used to mean both thumb support and thumb brace, you can safely ignore the differences. Just look at the features of the thumb support to find the right one for your condition.