Gauging the difference between classical and instrumental music comes down to the definition of either form, and it can be difficult to state precisely. Classical music has several definitions. It can refer to the music of a specific area or country, such as classical Romanian music, or to a strict period of Western music production between 1750-1820 that was expressed with a range of different instruments, instrument arrangement, and vocal productions. Composers of the classical music period include Mozart and Beethoven.
Sometimes, people use "classical music" to refer to any music written prior to the early 20th century that might be performed by an orchestra or a symphonic band. Classical music could be operas, solos, and small arrangements for chamber orchestras, quartets, quintets or trios. In this case, the difference between classical and instrumental music is the instruments: electric guitars, synthesizers, or any instruments developed after the early 20th century would be examples of those capable of producing instrumental but not classical music.
When comparing the two, it can be helpful to look at instrumental music that is produced in the style of classical music but written after the 20th century. In this case, the difference is based on the era. Some modern film scores, for example, are instrumental music, evocative of the classical era. Here, the modern film's score may sound like classical music, but isn’t because it was written recently.
Instrumental music in strictest definition is music played on an instrument. Some people include the human voice as an instrument, but others do not, so only classical music without singing is instrumental. Even if voice is included in the definition of instrumental music, many modern bands still would not be classed as instrumental. Further, they wouldn’t be considered classical, especially if they played in different genres like rock or ska.
When instrumental means without vocal accompaniment, any music composed without vocals would be instrumental. In this case, the difference between classical and instrumental music comes down to fine details and how strict the definition is of either form. To some, all classical music, vocal or not, is instrumental. To others, instrumental music means without vocals, so only some classical music qualifies. To others there is no difference between them, since both genres are produced on instruments.