To steal is more general. It can be in reference to any kind of theft. In slang, it can also refer to a really good bargain "The new car was a steal!" as in not expensive at all. To burgle specifically refers to stealing (something) from someone's house, car, or store. The noun form is "Burglary" and the one who burgles is a "Burglar." For example: "His house was burgled, the robber stole valuable items from him"

Steal is much more common and is useful in many different contexts. You can steal an apple from a supermarket, you can steal a car that is parked on the street, you can steal an idea from a competitor, etc. Notice that the structure is steal + thing. Burgle means to break into a place and steal things from it. We say burgle + place, e.g., someone burgled our house while we were away on vacation. You can't usually say "someone burgled my phone" or "he burgled an apple from the store." Another difference is that "burgle" is mostly used in British English. In the US we would say "burglarize" or "break into."