A "road" is a passageway for vehicles that goes from one place to another.
A street is a more urban concept. Streets are constructed by municipalities. Every street is also a road, but not every road is a street. For instance, it would be a bit odd to call a road from one city to another a "street". Odd, but not wrong.
Streets are often numbered: "23rd street", etc. This fits in with the fact that they are constructed by municipalities.