着火 = caught [on] fire whereas the other means burning.

It is hard for me to explain the difference in detail. But I can list some situations to identify which situation is suit for 着火, and which situation is suit for 燃烧. 1. If you are seeing a house burning, Then you call 911, you should say"这个房子--着火--了。" or "这个房子--烧着--了". If the officer ask you to describe the bad situation, you can say"情况很糟糕，整个房子都在--燃烧--。" 2. If you use a gas stove to cook. The first you should start the fire, when you see the fire come out of the stove, you can say "炉子--点着--了。" or “炉子--着火--了。” If the stove in a high quality, you can say "这个炉火--燃烧--的很旺。" 3. When you walk on the street, you are seeing a stranger's backpack is burning, you want to remind him/her, you can say "你的背包--着火--了。"

燃烧 is a more generic concept, and may without flame. 着火 must have flame, it is one type of 燃烧. 着火 is often triggered by an external fire source. 着火 often used to represent [be on fire].

it's like the difference between catch fire and combustion i guess

着火: catch fire; be on fire 燃烧: burn