For the US, oil money (in Dollars) and control over Iraq: https://www.middleeasteye.net/news/why-does-the-us-control-i...

For Israel, oil from Kurdistan: The revival of the Kirkuk-Haifa pipeline fell through but Turkey stepped in to enable transport.

For both, the neutralization of a regional actor. The chaos that also weakened its neighbors. Eventually, the destruction of Syria. A base and flight path to attack Iran.