Can you tell me the difference between a damaged tendon and a damaged ligament?

Yes. The tendons move your fingers and wrists. If it is damaged, you may lose the ability to bend or straighten the affected part. Ligaments support your joints, holding bones together. If that is damaged, you may have swelling to the region and instability to the joint. This means that it may hurt when you move the joint, or that it will move improperly.
Strain v Sprain. The definition of strain is damaged or torn tendon or muscle. The definition of sprain is damaged or torn ligament. Many people don't realize that. Ligament injuries are going to be joint injuries, whereas tendon injuries are often distant from joints.
Function. Ligaments join bone to bone usually to stabilize a joint. Tendons join muscle to bone. Both can be slow to heal when damaged and usually require a period of splinting. Both may require surgery to repair.