If you're delayed, they can book you a seat on a competitor's flight
Back in the golden age of flying, there was this thing called Rule 240, whereas an airline that delayed you significantly or canceled your flight was REQUIRED to rebook you at no extra cost, even on a competing airline. That ended with deregulation in 1978, but airlines will still do it if you ask nicely or if you have elite status.
Don’t expect the gate agent to scour the interwebs trying to find a seat for you, though. There’s likely 100 other people trying to get out as well, so if you make their job fast and easy you’ll get better results. Look up the flights you want, calmly stroll up to the counter with two or three options ready, and see if they can do anything for you. If those options include flights on their airline, all the better.
If your itinerary gets changed, they pay the difference...
If you’re massively delayed and the airline arranges alternate transportation with another carrier, they will cover all the expenses and extra fees the new airline might assess. So if there's only a first-class seat available, it's yours, and it won't cost you an extra penny. Pass the champagne.
... and, in that case, you get to keep your original ticket for later
That unused ticket for the delayed or canceled flight? It’s still good to use another time; think of it like an airline credit you got for your aggravation. If you’ve had it with that (expletive) airline and vowed never to fly them again, even for free -- you have principles, dammit! -- you can also request an “involuntary refund” for the flight you were bumped from.
Also, a point of warning: It HAS happened that airlines try and cancel your original ticket onsite, and confused passengers assume this is normal procedure. It’s not, so politely tell the reservations agent you do not want to cancel the existing reservation.