The primary difference is labelled right on your image of a TC: "NO PITCH INFORMATION". The artificial horizon is basically a pitch/bank position indicator, while the turn coordinator is a yaw/bank rate indicator. The horizon thus shows you how your aircraft is oriented right this instant, while the TC shows you how fast your orientation and/or heading is changing over time.
First, a correction from some other answers. A Turn Coordinator is not the same instrument as a Turn/Slip Indicator. They show similar information and are used interchangeably in normal flight, but they indicate slightly different things.
This is a Turn/Slip Indicator:
Notice the much simpler display. When the bar on the needle lines up with either of the home-plate symbols, you are yawing at a two-minute turn rate. No confusion with bank angle or any other indication of attitude.
The gyroscope in this device is basically rotating in a plane parallel to the aircraft's wings, and precession of this gyroscope induced by the aircraft turning is what drives the needle. However, a pure banking maneuver to start a "coordinated" turn doesn't show on the gauge; the aircraft has to start rotating relative to its up/down axis. This can make for a rough ride when attempting to bank to begin a turn, especially if that turn is being initiated by an autopilot; there's no gauge on board the aircraft to measure roll rate.
To solve that problem, the gyroscope in the instrument was canted 30 degrees in the plane from nose to tail; this makes the gyro sensitive to wing roll rate as well as fuselage yaw rate. This new version, the "turn coordinator" allows autopilots (and human pilots) to execute smoother turns in IMC.
However, there's a downside. Back in the 1920s, flying into a cloud, especially a large one, was often fatal; there was no artificial horizon yet, so once the pilot could no longer see the real horizon, if the plane's attitude were upset he'd be unable to recover. Then a pilot, Howard Stark, worked out a simple method using only the TSI and altimeter to recover from an upset; use the rudder to center the yaw needle, then ailerons to center the ball (with neutral yaw, the ball indicates the direction of gravity), and finally level out of any climb or dive until the altimeter is steady. This became known as the Stark 1-2-3 method and it saved a lot of lives.
The new "turn coordinator", because it responds to bank as well as yaw, will show you "yawing" again once you proceed to "step 2" and try to level the wings. Therefore, you can't use the Stark method with a Turn Coordinator. To indicate this and thus prevent pilots trying to use it as such, the display of the Turn Coordinator was changed to the symbolic aircraft:
The display makes some sense as the gauge will respond to roll rate, so it can be thought of as a combination roll/yaw rate indicator.
The modern method to keep an airplane in stable flight in IMC instead involves the artificial horizon:
Using a gyroscope mounted in gimbals, calibrated and subtly corrected in flight to rotate parallel to the ground (on an axis parallel to gravity while in level flight), it becomes possible to determine your exact angle of pitch and bank using a simplified representation of the real horizon. Blue is sky, brown is ground. Pull up, you see more sky, less ground. Nose down, just the opposite. Bank left, horizon rotates right, and vice versa, all like the real horizon out the window would be doing if you could see it. The instrument, as you can see, has markings indicating definite angles of pitch and bank.
Now, to recover from a minor upset in IMC, you simply pitch and bank to return the horizon to level, then while the horizon is stable, the Turn Coordinator will show you if you're yawing and how to correct it. This reduces the need for a pure yaw indicator in most normal situations. Extreme maneuvers, however, can render the AH unusable, so specialized "spin recovery training" is required to teach pilots how to recognize and quickly recover from extreme instabilities such as spins (uncoordinated stalls) and spiral dives.