Both terms refer to reductions in your settlement. "Betterment" means that your repaired vehicle is better than it was before it was damaged. Your insurance company can only charge you for betterment if the repairs increase your vehicle's resale value over what it was before the accident.
'Depreciation' refers to a reduction in your settlement based on the age or use of a part that is to be replaced. Certain parts on your vehicle have a 'life expectancy' and your insurance company may take this into consideration. For example, if a tire on your vehicle is expected to last 60,000 miles, but it was used for 30,000 miles at the time of the accident, your insurance company may elect to pay only 50% for a new tire. The company can also deduct the difference in cost if you insist on using a new part, rather than an available used part, appropriate for the age and condition of the vehicle.