Question: Do the same regulations that apply to drones apply to balloon mapping?

The rules for flying drones are different from the rules for flying kites or balloons. The rule that flying is prohibited within five miles of an airport applies to drones and balloons and kites, except that special permission has generally been granted to fly kites or balloons near airports. To get the permission, you must request that a notice to airmen (NOTAM) be filed. There is more detail here. I don't know if that information is up to date.

Chris

Also keep in mind that the rules that apply to balloons and kites have a lot to do with the altitude you plan on flying, but there are definitely differences between kites and balloons (which are considered moored objects) and drones, (which are considered as unmanned aircraft), from a regulatory standpoint, though there are additional considerations in regards to size, altitude, etc (doing a low altitude flight over a yard or community garden is generally not subject to the same kind of oversight as, say, putting a kite 1,000 feet up in the vicinity of a runway would be).

In addition to the link that cfastie provided, you might want to look here https://www.faa.gov/documentLibrary/media/Advisory_Circular/150-5200-28F.pdf to see more info about coordinating higher altitude kite or balloon flighting with local airspace regulators. : and here: https://www.faa.gov/uas/ for info about where drones may be flown (be advised that your city or municipality may have additional regulations: where I live in NYC they are illegal to fly regardless of airport proximity, though in practice there are plenty of casual drone operators who are either unaware or unconcerned by that fact).