If the flight price dropped after booking, the honest answer is that you can sometimes get the difference back, but not because the airline feels bad for you. It happens only through the fare rules you already agreed to, and those rules decide everything. There's no customer-service magic word. There's just what your ticket allows, and whether the new, lower price is enough to clear the cost of moving to it.
The two ways it actually works
The first is the 24-hour window. If you're still inside a day of buying and the price drops, you cancel for a full refund and rebook at the lower fare. That's the cleanest path, and it's worth knowing cold, because it's the one case where you get the actual money back with no fee and no credit. We wrote the whole thing up in the 24-hour cancellation rule: it covers most flights touching the US booked at least a week out.
The second is a change on a non-basic fare. Many standard economy tickets let you change your flight for no fee, and "changing" to the same flight at a now-lower price leaves the difference sitting as a credit on your account. You don't pocket cash, but you get travel value you can use later. The catch is that this almost never nets out in your favor by a few dollars. A $30 drop isn't worth a change if the fare difference, once the airline recalculates it, eats most of that.
Where the door is closed
Basic economy is where this falls apart, by design. Those fares don't allow changes or cancellations for credit, so a price drop the next day does nothing for you no matter how big it is. That's part of what you give up for the cheaper basic economy fare, and it's a real reason to price the standard fare before you book the lowest line.
So why watch a fare after you've already bought it? Because the next trip is the one you can act on. Seeing the price you paid drift down teaches you how much a route moves, and fares swing more inside a single week than most people expect. When we send the three morning fares, the point is to put you in front of a good number before you commit, so you're not the one refreshing the page afterward, hoping it falls.