Every time I search for "aeronautical engineering" on Google I see a few results about aerospace engineering. So they must be similar, but what is different about these careers?
A space engineer deals with objects in space: from satellites, space stations, space vehicles and space probes to space debris. Space means high launch vibrations, vacuum, radiation concerns, high temperature gradients and extremes, complex trajectories, next to no maintenance (meaning: built in redundancy)...
An aeronautical engineer deals with objects in the air, but not in space (space is commonly considered to start at 100km of altitude): mainly all types of airplanes and helicopters, but balloons and missiles fall under that category too. Aeronautics generally means cost effectiveness (mass produced), challenging mechanisms (in terms of performance and reliability over time, may mean redundancy), complex computational fluid dynamics, high stresses on the wings...
"Aerospace" is a mix of both. Since they are so different, I wouldn't say an aerospace engineer is worth a space engineer and an aeronautical engineer. Different fields, different skills, but such engineers particularly come in handy for the launchers, especially the new generation of renewable ones that fly (e.g. Virgin Galactic SpaceshipTwo, or Skylon).
Source: I'm a space engineer specialised in mechatronics. I have never worked in the aeronautics industry.
Technically, aeronautical engineers only deal with things inside Earth's atmosphere, while aerospace engineers may also deal with spaceflight. However, in practice the two terms are essentially interchangeable - you will want to look at each job offer / university course etc in detail to see what's being offered or required.