What makes tones sufficiently useful that it would lead to the evolution of tonal processing?
I think the whole issue is that you put all your focus into music. The ability to tell apart different tones hasn't evolved with the culture of music playing. It is much older.

Ability to distinguish different tones in non-human animals

Humans are of course not the only animals that are able to recognize tones. Mice for example can as well accurately tell the difference between two close tones (ref.).

Perceiving its environment

So why is the ability to tell different tones apart important. It is important because it allows one organism to better perceive the elements in its surrounding. For example, consider being in the savanna and you can't tell the difference between a lion roaring and a water fall. It seems to be a pretty important handicap.

Tones recognition in social contexts

A very large part of animal's communication is based on tons. Very primitive forms of language are mostly based on tones. Courtship display for example is often based on the tone of the other gender (the tone is correlated with the size of the organ producing the tone). A very large part of human's communication is based on tone as well. You might want to have a look at Albert Mehrabian.

If you have more questions about the importance of non-verbal communication you will need to ask your questions to psychologists (on CognitiveScience.SE) and not to biologists. Note btw, that the condition of being unable of recognizing different tones is called tone deafness.