What is the difference between grief and trauma?

Grief is a response to a loss. Trauma is an extreme fright. Post Traumatic Stress Disorder is when the extreme fright causes the victim to remain in a heightened state of vigilance to defend themselves. There are therapies to help with both. Grief can ease as a person becomes used to the loss. PTSD can be effectively managed through therapies designed to help with it.

Trauma is damage, and could be emotional.
Grief is response to loss as @Hawaii_Jake well put, so could be a response to the damage/loss of a trauma.

Trauma can cause grief, an many other factors can cause grief. Grief is a response, Trauma is a cause.