When is one processor more powerful than another?
When one processor can compute something that the other processor cannot.
But that answer is somewhat vague.
The idea of computer "power" is somewhat vague. Sometimes people use it to mean "computer speed," sometimes to mean "computer throughput," sometimes to mean "multimedia performance," and sometimes to mean "what a processor can compute." Often it is a fuzzy combination of all of these. Let us use the following definition:
Algorithmic Computing Power: Two processors have the same computing power if they can be used to implement the same algorithms (using each processor's machine language) and produce the same results.
For example, say that two processors have the same power. Then if a particular algorithm can be implemented on one processor then that algorithm can be implemented on the other processor, and both processors will produce the same result. This must be true for all algorithms (after implementing the algorithm in the machine language of each processor).
For another example, say you had a quantum computer Q and a conventional computer C. If computer Q could implement an algorithm that factored large integers in a small amount of time, but computer C could not be used to implement that algorithm then computer Q has more algorithmic power than computer C.
Processor speed is left out of the definition. As long as two processors eventually produce the same result with the same algorithms the processors have the same power. It is helpful to regard computing power and processor speed as separate aspects. Memory and peripherals (such as graphics boards) are also left out.
Can a processor that uses 64-bit integers compute things that a 32-bit processor can not?