two things (or people) that are incompatible and do not go together (like oil and water).
あの二人は性格が水と油で、長い付き合いは難しい。
Those two have personalities like oil and water; it's hard for them to have a long relationship.
伝統的なやり方と新しい手法は水と油のように感じられる。
The traditional method and the new approach feel like oil and water.
チーム内で彼と彼女は水と油で、いつも意見が対立する。
He and she are like oil and water within the team; their opinions always clash.