Architectural design is basically just an architect. It's a mash-up of art and science, but most college programs lean more towards the art side of things. You will learn to make cool looking buildings, and learn to make functional spaces. It's more than I can describe here, as there are volumes of books on the subject, but basically, google "Architect" and you'll see what architectural design is.
As for Architectural Engineering, that's what I do. A building is made up of thousands of different components, and all of them need to be designed. An architect might want a big grey stone wall there, but how is that wall going to stand up? If there is a light on that wall, how does it get power and how is it going to turn on and off? If you build a big glass box (architects love big glass boxes...) how are you going to air condition it so the people inside don't burn up when the sun's shining? There are a bunch of distinct majors involved in the building process: Structural Engineering, Mechanical Engineering (which is air conditioning, heating, plumbing, and fire protection), Electrical Engineering (which will be power, lights, and often telecom is included), etc.
I majored in mechanical engineering, so I learned how to design engines, aircraft, etc. None of that is needed for buildings. Likewise for other engineers. We learn all sorts of specialized things that might have nothing to do with buildings. An architectural engineering degree is specific to buildings and construction.
All architectural engineering programs will teach you the basics of architectural design, structural engineering, mechanical engineering, and electrical engineering, as it pertains to buildings. But most programs will emphasize mechanical engineering (and a few may have electives to emphasize structural).
If you've got something specific in mind that you like more, you'll need to read up about different school's programs to see which architectural engineering program is right for you.