Live recording is recording a whole band performing at the same time, while multi-track allows individual band members to record their own tracks. A guitarist could want to add multiple parts to the song to be layered or played in different parts with different tones, multi-track recording allows this. It allows much more creative freedom.
A result of recording the band all together is that you can’t really add effects to individual parts and it’s very difficult to edit. However, recording a band this way would add to the authentic sound and make it sound more natural. Live recording works well for jazz as the feeling of recording a single part in jazz is alienating and the emotion put into the performance would be reduced.
Dodie’s Lonely Bones recording is a good example of multitrack recording. She layers each instrument and vocals at different times with different people all either in a studio or at home. This results in lovely vocal harmonies recorded to sound like a crowd.
On the other hand, the tiny desk performance dodie and her band did is an example of a live recording as it’s a live performance.