Short answer:
hell yes you got a faster processor.
Hertz is literally a measure of
cycles per second; in terms of CPU's and GPU's, it's how many clock cycles a processor will go through per second. It's what you do with those clock cycles that matter.
Let's think of it a little bit like RPM's in an engine -- if you're hauling a big heavy trailer, which is faster, 4000 RPM's or 8000 RPM's?
In reality, there's no answer to that question -- RPM does not dictate
power. A Honda Civic Si 1.6L will make it's peak power at 8000RPM's of around 160hp, but the Peterbuilt 8.3L Turbodiesel next to it will make it's peak power at ~2500RPM's of about 450HP. Ultimately, RPM doesn't mean anything unless you know what the "engine" can do with it. Similarly, trying to estimate a CPU's ability to get things done purely by Ghz (billions of hertz) really has no basis in reality.
The "netburst" architecture (your old P4 2.4Ghz) had a very low "IPC", or instructions-per-clockcycle ratio. Even though the speed could get very high, it would take many
many clock cycles to get something done...
The "conroe" architecture (your current E6600 2.4Ghz) has a massively improved IPC; some 2x better (or more, depending on the circumstances) than your old P4 processor at the same speed. Thus, at equal clockspeeds, you should expect your E6600 to be somewhere around 2x faster than your P4 processor on single-threaded applications.
Another big benefit comes in those applications that can use multiple processors simultaneously -- most games and operating systems have finally now come to the point where they can use dual CPU's. In these circumstances, you'll see even further benefit -- an average of ~50% improvement on top of the original 2x; some apps can see more, others may see less.
Want the icing on the cake now? The Core series of processors can do all this while using about half of the power of your old P4. That means less heat, less fan noise, less stress on your motherboard and PSU, and smaller electric bills.