The main reason cache is important is that it increases the real speed of a processor by providing the processor with data more quickly. A processor can only crank through data if it is being given data, and any delay that exists between when the processor requests data and when the processor receives it means that clock cycles are being left idle which could have otherwise been used.
That said, summing up the importance of cache in a general way is difficult to do. Cache performance has traditionally been ignored because until very recently it did not have a major impact. The cache for most Pentium 4 series processors, for example, was only 256 or 512 kilobytes, depending on the processor. Having some cache was useful, but it did not have major impact on application performance because there was not enough cache for applications to make use of. For example, the original Celeron had no cache integrated into the processor, but it still performed extremely well in games compared to processors which did include cache. This was partly because cache sizes were not large enough to hold more than extremely common and general sets of data and partly because games generally don't make much use of CPU cache because there the amount of frequently used data far exceeds what is available.
However, cache has become more important in modern processors thanks to better implementations and an increase in cache size. This increase in the important of cache is probably due to the problems that became evident in Intel's Pentium 4 line-up. The Pentium 4 processors relied heavily on high clock speeds, and as a result they were power-hungry and ran hot. Intel learned from this when it created the Core 2 Duo series, which used lower clock speeds but made up for it with multiple cores and a much larger cache. These days, it is not unusual to find a processor with an L2 cache of 8MB. Cache has become important because it gives Intel and AMD a way to increase the performance of their processors. Cache goes hand-in-hand with the trend towards processors with multiple cores. Both the Phenom II and the Core i7 possess larger cache sizes then the Phenom and the Core 2 architectures, and the trend of increasing cache size is likely to continue.