For the uninitiated: MIDI is a fundamentally different ideology of storing sound compared to MP3, AAC, and so on. Whereas MP3 and their brethren store actual sound data, MIDI stores what is most easily described as musical scores.
When you play a MIDI file, you are expected to literally provide the musical instruments to play the musical score with. In the past this was provided by your sound card with a synthesizer, and in more recent times provided by your operating system as a software synthesizer.
The reason behind MIDI being structured this way was to reduce file size. Remember, we are talking the 1980s when MIDI was invented; disk space was expensive. Storing the raw sound data requires a lot of space, but MIDI only needs to store the musical scores which are significantly smaller. The actual sounds were stored locally and synthesized by the sound card.
In an era where disk space is worth pennies, MIDI is an obsolete solution for a problem of a bygone age. But the technical considerations and compromises that went into its design and the innovative creations that composers made within the technical restrictions imposed are nonetheless a hallmark of computing history.
MIDI was designed as a way to link up a controller like a piano keyboard to a synthesizer and transmit the notes being played. In that area it's certainly not obsolete: synthesizers still transmit and receive MIDI messages and sequencer software still records them. It's the only sane way to edit and view note data because you can't edit audio easily to change a few notes.
As a way of transmitting musical scores it's obsolete, but mostly because it was really hard to get MIDI soundtracks to sound as intended because there were so many different synthesizer implementations.
I'd say the openness of the format deserves more recognition and is far from obsolete. Its rightful successor is tracker music, which includes the instrument samples along with the musical score. Pure waveform is a lot like closed-source software, with many inherent drawbacks.