What's the difference between "to get old" and "to get older"?
Not a teacher only a native.
I would explain the difference like so:
To get old - suggests that a person has reached old age eg. 70 or 80.
To get older - simply means growing up, progressing through the years.
For example someone could be 13 and said to be 'getting older' due to them aging, or perhaps maturing in attitude.
Thats how I see it anyway.
I agree with Shroob's distinction, but would add that the term older people is favoured by many instead of elderly or old, so you'll hear get older being used as a polite or euphemistic way of saying get old.