Though there appears to be a lot of similarity between the two water rides, wakeboarding and water skiing are two completely different sports.
The nature of both the rides is a key point of difference between the two. Water skiing is a simple sport whereas wake boarding seems to be a combination of different sports. The principle of water skiing is based on being towed around by a speed boat. Though wakeboarding follows the same principle, it is more of a combination of surfing and snow boarding along with water skiing and a little bit of skate boarding as well.
Water skiing traditionally is just a ride behind a speed boat. This is pursued for the simple pleasure of skiing over waters. A water skier prefers smooth sailing over the waters. Wakeboarding is an offshoot of surfing which means that the wake boarders prefer larger waves to ride on.
The different nature of the two sports is reflected in the individual sports gear as well. Skiers use simple water skis while wake boarders use boards similar to surfing boards but with fins and foot straps included for better balance. Unless you are slalom skiing, skiers generally use two different skis for both feet whereas wake boarders use a single board.
The speed boats used for these purposes also differ from each other. While the water skier may use any simple motor boat for skiing, the wakeboard boat is specially designed to create extra high waves as the thrill of wakeboarding (as in surfing) is to ride these waves without loosing one's balance. The boats used for wakeboards have increased weight at the rear to create higher and steeper waves. Almost all wakeboard boats have the engine placed backwards at the rear of the boat as a load increasing measure. The lower the stern the higher the waves produced. Newer technologies are being used as manufacturers compete with each other to create boats with higher waves.
Ropes used in wake boarding again differ from the ones used in water skiing. While the ropes used in wakeboard belong to the non stretch variety, the water ski sports requires ropes with some stretch which helps to pull the skier back and forth as he rides along the curve.
OK, so much for the differences between water ski and wakeboarding.