What is the difference between azimuth and heading?
A heading (in the general case of moving "forward") is the direction your nose is pointed in. This may not be your course (as discussed here).
Headings are measured from onboard a traveling vehicle or object (e.g. from the cockpit of an aircraft or the bridge of a ship at sea).

An azimuth is a bearing, more precisely a compass bearing from a specific point of observation like a radar station. (The "North" used as a reference may be either magnetic or true depending on the system you're working with, but for purposes of this discussion it doesn't matter.)
Often the point of observation is fixed (ground radar, a control tower, an artillery spotter, etc.), though it need not be as long as the observation can be mapped to a compass bearing somehow. 



Bonus Answer: The "O'Clock" positions are an informal type of relative bearing as mins noted: The 360-degree circle is broken up into 12 chunks (30 degrees each) for easier scanning, with "12 O'Clock" being directly in front of you and "6 O'Clock" directly behind.  

An important caveat to be aware of is that when ATC provides a "clock bearing" for traffic it's relative to the aircraft's course (12 O'Clock is aligned with the aircraft's track over the ground as shown on the radar). Depending on wind there may be considerable variation between your course (direction of travel) and your heading (direction the nose is pointing), so the called traffic may appear in a different location when you look out your window.