I believe that I can speak with a certain amount of experience on this subject, having been a Tarot/Rune reader for many years during my neopagan phase. I still have my cards and drag them out every now and again. They're anything but a game, in my opinion, but let me explain why before anyone jumps in.
First thing I will say here is: pay attention to athon. He's a really smart guy, and he's hit the nail right on the head from his first post in this thread.
Now.
The "power" of Tarot (such as it is) is the power of symbolism. I like to use a deck with detailed illustrations, such as the Rider-Waite, or my personal favourite Norse Tarot. Each card has a situation illustrated on it, and when you draw these out randomly, you make connections between them. That's what people do - they make patterns.
Each card has a carefully-constructed story associated with it - or rather, part of a story. You need very little assistance to link these independent story pieces together, and even less to relate it to something in your own life - especially if you approach the subject with an a priori assumption that it is going to relate to your life, which each and every querent does.
It has been my experience that the stories generated randomly by the Tarot have a tendency to illuminate, or clarify, someone's thinking on a subject. They're a assistant to decision making. Let me use an analogy.
When you flip a coin to make a decision, how many possible outcomes are there? Two? Wrong.
There are actually four possible outcomes to a coin-flip-assisted decision: a heads you like, a heads you don't like, a tails you like, and a tails you don't like. Flipping the coin helps you to decide what option you prefer by "forcing" you to take one or the other. How many times have you said "heads hamburger, tails salad", flipped heads and decided "you know what? I think I'll just have a salad," contrary to the result of the flip?
Tarot works the same way. By putting symbols in your way, you can identify options and make decisions that you otherwise would vacillate over, or be ambivalent about.
Tarot, like a coin flip, is a decision-making tool. Should I quit my job because I hate my boss? Nine swords suggests that you're overreacting to the situation. Five coins indicates possible financial difficulties. The Sun shows that there is a brighter future ahead. Okay, I won't quit my job.
Can the Tarot predict future events? It depends on what your definition of predict is. Certainly it can form part of a self-fulfilling prophecy. I didn't quit my job because of what the cards said, and now the boss I hate has retired. Did the cards predict that? It depends on your interpretation. Certainly the "brighter future" would not have happened if I had ignored the cards and quit my job, and then I wouldn't have thought twice about it. But the fact that humans are pattern-makers, and the well-known phenomenon of confirmation bias, means that I will attribute more meaning to the cards than is inherent in the images alone.
- I already know. I just drew The Tower card).