Composing Written Music
In the simplest terms possible, composing is the act of writing a song. Composing is the most vital part of music, and while most musicians are able to read the music written by a composer, few actually believe themselves to be proficient in composing. It's a tricky form, especially when a composer is given the task of composing for an entire orchestra or band as opposed to simply one instrument, such as individual piano or cello compositions.
Strictly speaking, the terms composing and songwriting are completely interchangeable. They do, after all, mean exactly the same thing: writing a song, or a part of a song, from start to finish. But the words' connotations are radically different. While songwriting is an informal practice, confined mostly to the popular musical genres, composing is the province of the classical musician. Those interested in composing must not only be able to perfectly read music, they must be able to notate it equally as well. Songwriting, while a valuable form in its own right, is rarely used as a means to write down music for others to read; those writing the songs are frequently writing for themselves or for others to whom they can simply play the song. Composing, on the other hand, is usually intended for an individual other than the composer -- and for widespread consumption, at that. Great compositions will migrate from orchestra to orchestra, eliminating the possibility of the person composing the music to physically play the song for other musicians.
Composing often requires formal training (though it in no way has to; several great composers have had no formal training at all) and many would-be composers attend music school for years prior to the start of their careers. A music school focused on composing will educate its students in massive amounts of music theory as well as individual instruments. Most composing students are encouraged, if not required, to study and proficiently play more than one or two instruments aside from piano, which is a given.