If it is a non-refundable fare, they often charge you a fee to change the ticket. For example, I purchased a $124 one-way ticket from Akron to Orlando for my wife. Even though the price later dropped to $79, I would have had to pay $50 to re-ticket. So it wasn't worth it to me to make the change.
If the airline makes any changes to your flight (departure or arrival time) from when you purchased the ticket, you should be able to cancel the ticket and get a full refund. You could then simply repurchase the ticket at the reduced price. Of course, convincing the airline to give you your money back is always a challenge.