A symphony is a large multi-movement composition written for a full orchestra. The ensemble is the main focus: themes are shared across the orchestra, and Classical-era symphonies commonly use about four movements (often with a fast first movement in sonata form).
A concerto is also orchestral and usually multi-movement, but it sets one or more solo instruments against the orchestra. Soloist and ensemble relate by alternation, competition, and combination, often highlighting virtuosity; since about 1750, concertos typically use a three-movement fast–slow–fast plan. (Encyclopaedia Britannica)