hurry: to go somewhere or do something more quickly than usual, for example because you are late or you must finish something soon
If you don’t hurry, you’ll miss the bus.
We have plenty of time. There’s no need to hurry.
rush: to go somewhere very quickly, or to do something too quickly and without thinking carefully enough
Everyone rushed out into the street to see what was happening.
Try to answer the questions calmly, without rushing.
A police car rushed past.