Tone and mood are closely tied to one another. Here's a simple explanation on the difference. Tone would be the writer's attitude toward his/her subject matter. Mood is the audience's feelings as they read that particular piece.
I would say that the tone of "The Outsiders" is realistic. Hinton doesn't shy away from the brutality of the Greaser life. It reads like she wanted to shine a light on a subject that people were averse to seeing. In some ways it reminds me of investigative journalism. The point is to shed light on a social injustice AND tell a story at the same time. You can't say that Hinton's tone is cold and unattached. She isn't just reporting the facts. If that were the case, there would be a lot more narration on the Socs. Hinton's tone is focused on the plight of the Greasers and she wants her audience to know each facet of that Greaser life.
The mood changes throughout the novel. In the story the mood is a bit rebellious. It's clear that Hinton wants the reader to identify with the Greasers and feel that disdain toward the Socs and their money. That rebellious mood turns toward anger. The story about Johnny getting jumped by 4 Socs helps sell that mood. Fear is present in the story as well. For example when Pony and Johnny are on the run and hiding away in the church. Sadness is the mood Hinton establishes when Dally and Johnny are both killed. All in all, the mood of "The Outsiders" is not happy. It ends a bit hopeful, but is never happy.
the mood at some points is also dark and mysterious like when they're at the park