Why do files have different formats?

Answer 1

Files exist in different formats in order to provide for any task a suited format that does what it is used for as close to optimally as possible. Sound formats store sound information, image formats store image information, and you can't use an image format to store sound information.

But even inside these categories there are different formats. Some are just older formats that are still used despite there being a better suited format available.
For example, the BMP image format could be called obsolete, because the JPG format achieves what it does better - it stores images at a lower filesize with equal quality. However, BMP has an advantage over JPG in a certain field, which is universality. That's why BMP is still used sometimes today.

It's very similar in the other file categories. There are many types of sound file formats, and each of them has its advantage over the others.

One frequent mistake is mixing up "file format" and "file extension". While one file might be called ".2p3" and another one ".mtf", they could have the same file format as ".txt" - plain text. Software can create files with different extensions as the developer needs it, but the file format is usually the same as an existing one.