Will a new processor with slower clockspeed run legacy applications faster?

The clock speed is only partially in charge of 'how much stuff gets done' (a.k.a. 'work'). To compare 2 CPUs you should check out CPUbenchmark.net and find both your old and your new CPU and compare the results.

For example, a P4 with 3 GHz clock speed achieves a benchmark result of 491 (see here). the Core 2 Duo with 1.8 GHz clock speed achieves a result of 1115 (see here).

So, the Core 2 Duo is able to do more work in the same amount of time. With that in mind your drawing app should do calculations faster, and it will wait for your input much faster :) (it idles more since it finishes the tasks faster).

As been pointed out already, clock rate has very little to do with how well a CPU performs. It all comes down to how many clock cycles that is necessary to execute a given instruction. This is called CPI (Cycles Per Instruction), and it's what measures a CPUs performance.

So even though the Pentium 4 has a much higher clock rate it will be greatly outperformed by the much newer and more advanced Core2Duo.

If you want to crunch some numbers you should check out CPU Worlds benchmarking database.