Magento 1.9.3 CE, Staging & Live Sites behave differently

Lately, we found our Staging Site and Live Site behave differently in 2 situations: 1). we found Live Site to have 500 error timeout while Staging Site is normal; 2). we found update on CSS file in staging site reflected in frontend, while Live Site do not reflect the changes.

how to check the code difference, and how to sync the Staging site code to Live site?