In terms of hardware, the Pro features an OLED touchbar at the top of its keyboard rather than a row of function keys. The Pro also comes with 8 GPU cores by default. The Air only has 7 GPU cores unless specially configured. The difference in cores should only have a small impact on graphics in most contexts.
Internally, the Pro has a fan that provides active cooling to the CPU and GPU cores, allowing it to sustain high levels of performance for longer than the Air. And although both are thin-and-light machines, the Air is going to be somewhat lighter. The Air's body has a wedge shape that helps it feel thinner. The Pro will feel a bit blockier, and its fan will create some noise when the system is dealing with heavy performance loads.
The Pro's screen maxes out at 500 nits of brightness, 25% brighter than the Air's, but both screens are similar in terms of color gamut and accuracy. The Pro has a larger battery, which should deliver longer endurance than the Air when performing similar tasks. Battery life for all machines like this will depend heavily on what they're being used for, however. If you're performing demanding tasks, it may be that the Air actually lasts a bit longer than the Pro, because the Air will throttle performance more. The Pro will engage its active fan to stay faster for longer.
Both computers run identical MacOS versions, so the only software differences will most likely relate to the OLED touchbar on the Pro. Software is used to configure what appears in the touchbar, while the Air always has the same set of hardware function keys at the top of its keyboard.
Overall, the Pro seems designed for people who want a thin-and-light machine that can tap into the best of its hardware performance for sustained periods of time. The Air makes a small sacrifice in sustained performance in exchange for an even thinner and lighter design along with silent operation at all times.