[ The Dictionary of Musical Terms, publ. by G. Schirmer, 23rd ed.,
 [ 1923, gives these definitions:
 [
 [ arrangement: The adaptation of a composition for performance on an
 [ instrument, or by any vocal or instrumental combination, for which
 [ it was not originally intended; hence, the composition as so adapted
 [ or arranged.
 [
 [ transcription: The arrangement or adaptation of a composition for
 [ some voice or instrument other than that for which it was originally
 [ intended.
 [
 [ Yep, they're interchangeable!  -- Robbie

Robbie's comment is perfectly accurate as it relates to these two
dictionary definitions, but I believe there is a further, wider
definition of "arrangement", and a somewhat narrower definition of
"transcription" which are not addressed here.

Here is my understanding of the two terms, from current professional
usage:

Transcription:
This is the more academic of the two terms and relates to the literal
adaptation of an original composition for another performing medium
without any material alteration of the melody, harmony, counterpoint,
or the "nuts and bolts" of the music.

A good example would be the adaptation of a piece of classical
orchestral music for organ, orchestrion or piano solo, hand-played or
otherwise, retaining as much of the original musical detail as possible
but without noticeably re-arranging the notes.

Arrangement:
This is the more fun term.  You can do anything you like to the music
when re-arranging it: you can change the harmonies, rhythm, anything
you like; you can "jazz up" a classic, change a waltz to a two step;
the sky's the limit!

The kids now would call it "re-mixing" (although that is something else
again...).  Good examples would be any of Gustav Bruder's popular song
arrangements for the various Weber instruments, or Frank Milne's
Duo-Art arrangements

I think that the bottom line here is that "transcription" implies a
more academic approach to the music in cases where the source material
exists as a sole unimpeachable entity -- for example a piano piece by
Chopin or an orchestral score by Wagner -- while "arrangement" is a
looser term, usually but not exclusively used in relation to popular or
lighter music where the source material is often just a melody with a
simple harmonisation, intended by the composer to be further arranged
by another person.

Rowland Lee
UK