interconnects vs speaker cables vs power cords - which make the biggest difference?

As others have mentioned, it's system dependent.
With that said I'm going to answer your specific question and not get into other areas that you didn't ask about but others alluded to.
Power cords, interconnects and speaker cables in my opinion carry a signal or current of some type and can be used to tune your system (sound wise) to your liking. I find all three equally important.
If pressed to rank them in order, I'd agree with others that said:
1) Power cords
2) Interconnects
3) Speaker cables

Yep, Good quality Power Cables through some sort of Power conditioning 1st, then good quality interconnects 2nd. I just havn't been able to get big improvements with Speaker Cables so far, so a distant 3rd for me at this stage.

Factors that tend to increase the criticality of speaker cables:

-- Low speaker impedance.
-- Long cable length.

Factors that tend to increase the criticality of line-level analog interconnect cables:

-- High output impedance of the component driving the cable.
-- Use of unbalanced interconnections.
-- Long cable length.

Factors that tend to increase the criticality of power cords:

-- The component being powered has an unregulated power supply and draws large amounts of current that fluctuate widely. Namely most power amplifiers and integrated amplifiers.
-- The component being powered tends to generate significant amounts of electrical noise, that may couple via the power cord to other components. Namely power amplifiers, integrated amplifiers, and digital sources.
-- AC line voltage at the particular location that is lower than nominal.  ("Nominal" being 120 volts in the USA).

Phono cables and digital interconnect cables involve complex issues that I won’t address in this post, but their criticality will vary widely depending on the specific equipment that is involved.

Regards,
-- Al

IMHO the speaker cables are the most critical.  I'll throw in a plug for the brand I use, Jena  Cable which are simply insulated braided copper. Then ICs, I have yet to hear a meaningful difference between power cords.