A musical composition is creating an original piece. You create new musical material, chose the structure, harmony, rhythm, etc. Whereas an arrangement is a "re-coloring" of a previously composed piece for a different instrumentation.
For example, if you were asked to arrange a piano and vocal piece for orchestra, you would want to keep the musical themes, rhythms and harmonies as they are, or as close to the original as possible. (Harmony can be difficult and limited by what you are arranging for. Example is if you are arranging a symphonic work to be played on piano. Perhaps there are too many notes to play on piano, so you would need to decide which notes are the essential parts for either the moving line, harmony, etc.). A famous example of arrangement in classical music is Ravel's orchestral arrangement of Mussorgsky's Piano piece, Pictures at an Exhibition. That is a strict definition, and can be widened, but the essence of the piece must remain the same. In a looser context, arranging music is simply recognizing the piece from beginning to end. The loser application allows you to change harmony, rhythm, etc. in localized places to change the feeling (or emphasize) the feeling. An example of this is Jacob Collier's jazz arrangements.
The difficulty in differentiating arrangement and composition can occur when a composer utilizes the musical quote. A musical quote is when a composer uses a previously composed piece in part (a sample). This is a recontextualization of a musical idea within a new composition. Example: Ive's quote of Beethoven's 5th Symphony in his Piano Sonata, mvmnt. III. Another example is the Orchestral Hit in sampling. This can become a little confusing to some as composers, especially ones in the 20th century begin incorporating larger quotes, such as Gubaidulina's quote of Webern's arrangement in her violin concerto.
