What's the difference between 拍照 (pāizhào) and 照相 (zhàoxiàng) which both mean "to take a photo"?
The two terms have the same meaning of "take photograph" in modern Chinese, they just describe the same action differently.


'照' in '照相' refers to 'reflect with mirror' and 相 is short for 相片 (portrait/ photograph). Taking photograph with early cameras required the subject to stand still for few minutes, facing the lens like facing a mirror; even with modern cameras, the subject still need to stand still in front of the camera lens to have his or her picture taken.
'拍' in '拍照' refers to the 'clicking' sound of the camera. and '照' is short for '照片' (photograph). when you push the camera button, the shutter would open and shut and make a 'clicking' sound. 


(it might also refers to the loud flashing sound the early camera flashlight made)

照相 probably was coined before 拍照. In early days of cameras, the only way to take a photograph of yourself is  go to 照相館-- We never heard of '拍照館' And taking picture of a person or a family was the sole function of 照相館. 

Also, we have 禁止拍照 signs but not 禁止照相 signs. 禁止拍照 mainly targeted handheld cameras. With handheld cameras, we can take picture other than people, therefore, 拍照 can mean taking picture of people, scenery or object; While 照相 mainly means taking portrait of people.