Synthetic oil has a few benefits over conventional. First, it manages to hang on to its design properties longer... It resists break down for more miles.
Second, it outperforms conventional oil outside of the normal temperature ranges for an engine. In other words, if your engine goes above 195, it doesn't get too thin (good for endurance racing or hot lapping at the drag strip). If your engine is really cold (middle of winter) it doesn't get too thick.
Also, some tests try to prove that you pick up a few horsepower with some synthetics over using conventional. Royal Purple spews that stuff a lot, but frankly, that crap is just too far overpriced.
But in the end though, if you run conventional and you use a good filter and change it on regular intervals, your car will run for hundreds of thousands of miles with no problem. It's really more about changing it regularly and using a good filter than anything else.