Let me try to give a more interesting example, that can't be expressed as a set of simple regular expressions. BNF, properly speaking, is about the structure of a thing, such as a language. As such, the parts have relationships to one another. So, in computer languages "if" and "while" are tokens (word symbols, keywords), and tokens can be defined by regular expressions, the sequence "if while" isn't legal in Python, since the BNF that defines the structure contains no such form.
The following won't be perfect, and I'll need to leave out a lot for want of time to complete it. Take ellipsis (...) as meaning that I'm leaving things out, not as any formal symbol. Literals are quoted. Unquoted words are defined with other BNF productions in the same language (the non terminals of the BNF.
And note that I'm trying to generate a description of a car, not a car.
carDescription ::= chassis "containing both" propulsionSystem "and surrounded by" body.
chassis ::= ("rectangular" | "triangular") framework "with wheels at vertices".
framework ::= "reinforced" ( "steel" | "fiberglass") "semirigid structure".
propulsionSystem ::= fuelPoweredEngine | motor
propulsionSystem ::= (fuelPoweredEngine | motor) propulsionSystem
motor ::= "electric motor and battery"
etc.
Note that propulsion system is the recursive way of saying that a propulsion system is one or more "thingies" that provide power. It is equivalent to the + in regular expressions.
But the intent isn't to define individual, unrelated things, but the overall structure of a complex things that is made of parts, some of which are other complex things. Regular Expressions don't have that power, but BNF does.
And please note that my intention here isn't to be completely accurate (pedantic) but simply to try to demonstrate the essence of BNF as opposed to RE. It is one of the fundamental ideas of the Chomsky Language Hierarchy.
Since BNF and REs were actually created in the service of defining languages such as Python, it is difficult to divorce the idea from language. Hence my example is for a description of a car (which is language) not for a car itself, which is too complicated to express as BNF since the actual form of the connections between things in an actual car are critical. I left out the drive train, for example, which connect the propulsion system to the wheels and the control system that regulates the propulsion system. I don't have a theorem for you but am pretty sure that it would be beyond BNF to do this. Even if it were, the complete description would need the complexity of an engineering diagram and even then would leave out physical principles of combustion or conversion of electrical to mechanical energy.
