Your dad wants Mozart in the car and you want Eminem? That's an irreconcilable difference right there — two tastes or ideas or preferences so different from each other there can be no compatibility or middle ground. Diplomacy can't help, only headphones.
Where would divorce lawyers be without this lovely word irreconcilable? Their clients don't just have differences with their spouses, they have "irreconcilable differences" — in other words, demands or points of view so distant from each other they can never be bought together. Bye-bye marriage, hello legal fees.