Just what Jonathan said. The first one is focused on the actual machine/vehicle...an airplane ✈️...the "what". The second one is focused on the mode/means of transport...how you usually travel...using an airplane 🛫....the "how". Think of these two questions: Q1. What are you traveling on right now? A1. I am traveling on an airplane. Q2. How do you usually travel? A2. I travel by airplane. Hope this helps.

Both are correct, but there are shades of meaning. A noun without an article becomes the name of a class or category. "By airplane" means "by the METHOD of flying in one of those machines". There are many categories of transportation: by ship, by car, by plane, walking, etc. "Airplane" (with no article) is one of those CATEGORIES. "Airplane" is not a machine (even if "airplanes" are machines!) "An airplane" refers to an actual machine. ("The airplane" also refers to an actual machine, but to an even more specific one.) So "I travel by an airplane" means you will travel in one of those MACHINES. Are you confused yet? They seem the same, don't they? Indeed they are very close. The first specifies the METHOD or CATEGORY of the transportation. The second specifies the type of MACHINE. There is no factual difference. The difference is in your head. However, what is in your head matters a great deal. The purpose of language is as much to reveal your thoughts as it is to communicate facts. I can't tell you which way to say it. Let me ask you a question. Imagine you about to fly somewhere. Which comes to your mind first: the means of transportation or the machine that will take you there?