At first glance, skeleton and luge look very similar. Both sports involve people racing down the ice track on tiny sleds, the only obvious difference between them being that luge riders race on their backs, feet first, and skeleton riders race on their bellies, head first. However, despite the fact that skeleton is head first, luge is the more dangerous of the two, and there are several differences between the two sports.
Sleds
First is the sleds themselves. A skeleton sled rides on top of tubular runners, and on the body side of the sled, there is a saddle to hold the rider in place.
Bottom of skeleton sled:

Top of skeleton sled:

A luge sled rides on top of steel blades that are much sharper than the tube blades of a skeleton sled. This contributes to luge being faster than skeleton.
Luge sled:

Aerodynamics
Another reason luge is faster than skeleton is aerodynamics. The feet-first position is more aerodynamic than head-first, because the relatively flat front of the skeleton helmet introduces drag.
Steering
The steering is very different between the two sleds as well. In skeleton, the rider has a very good view of what is going on. The rider steers by flexing his body. Because of the saddle, they have a lot of body contact with the sled, and have a lot of control with small movements. When they nudge their head a little, their body responds, and the sled responds as well.
Luge is much harder to control. A luge rider lays on top of the sled, and there is no saddle holding them to the sled. Laying feet first, it is much harder to see where you are going, and there is less body contact due to the lack of a saddle. The runners of a luge sled are bowed up at the front and contact the riders legs. Steering is done by a combination of pushing on the runners with your legs and flexing the sled with your shoulders.
Luge also has a 2-man variety as well. Typically, the 2-man luge is slower because they start further down on the track, but it is much harder to steer because the man on top is the only one that can see, but the man on the bottom is the one that is most able to contribute to the steering.
Start
The races are started differently. In skeleton, the rider starts running with the sled at the top of the track and belly flops on to the sled. For luge, the rider starts sitting on the sled, and grabs handles attached to the track on either side. He pulls the handles to start the sled moving, then pushes the ice on either side of the sled at the start of the race using spiked gloves, before laying back and assuming the race position.
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