On the spot
I said 'hurt' and 'injure' are comparable in usage, but 'injure' seems more serious than 'hurt'.
'Damage' is used for things that are unrepairable naturally.
She hurt her good leg trying to support an old injury.
- not so serious
She injured her good leg trying to support an old injury.
- sounds serious, like she'll have a difficult time for a while
She damaged her good leg trying to support an old injury.
- sounds serious an the problem won't heal or the effect is lasting
Context plays a big part. The usage changes if we're talking about feelings or inanimate objects.