Recording Studio Definition
A Recording Studio is simply an environment to allow for the capture, manipulation and mastering of an auditory product (i.e. like a song).
The term Recording Studio should not be confused with the term Production Studio. Usually, after the recording studio is used to product an auditory product, it is then often moved to other (production) studios to be further edited and synchronized with other media. Examples of this "other" media would include Video and graphical production.
Here is a link to a more detailed definition of a Recording Studio (WikiPedia):
http://en.wikipedia.org/wiki/Recording_studio
Audio Engineering is the means by electronic equipment to record sound.
Here is a link to a the WikiPedia definition of Audio Engineering: