"I'm on my way to the airport" would be used when you are literally in the car and reporting to someone about it in real time, while in the process of going.
"I'm going to the airport" could be said for any time condition. You could say, "I am going to the airport now, or tomorrow, or next week." But we would never say, "I am on my way to the airport next week."
"On my way means" right now this instant. If someone is late they would call and clarify, "I am on my way." If they said "I'm coming" this would be a bit rude and not clarify that they are moving. "I'm coming" could mean they are still at home and eventually will leave." But "I am on my way" means they are rapidly approaching and will be there soon.