As ka9q has pointed out their would certainly been improvements in electronics, since most of the 60's Apollo electronics were built on RTL framework, stable but obsolete currently.
Not just RTL, but electromechanical relays. I haven't looked much at the Saturn V logic, but I've looked a lot at the CSM and LM. Although RTL is the basis of the AGC, it is hardly used elsewhere. Logic decisions are made mostly with combinations of mechanical switches, relays, diodes and discrete bipolar transistors. One-shots are frequently used for timing. (That's a circuit that, when activated, produces a pulse of a specified duration.)
Because switches and relays are mechanical and therefore unreliable, a lot of effort went into mitigating single failures. Several critical functions, e.g., SM Jettison, have two switches in parallel.
Nowadays these functions would be implemented totally differently. If I were designing the architecture, I'd dedicate a small microcontroller (or two or three, for redundancy) to each specific function, e.g., sequencing the parachutes or keeping the high gain antenna pointed at earth. Communication would be over shared or switched digital links, much like today's cars use Canbus,
greatly reducing the amount of wiring. There'd still be plenty of room for a redundant bus or two in case one gets broken or shorted. Communications between modules (e.g., SM/CM, CM/LM) could be optical to simplify and improve the reliability of the connections. That would leave only electrical power, oxygen, water and possibly coolant that would require separate lines in a SM/CM umbilical.
Many controls would move to touch screen, but I would spend a considerable amount on human factors research in which the astronauts would be closely involved. I think a lot of present-day touch screens are very badly designed because nobody really pays attention to how they're used. I also think they go too far in moving
everything to touch screens, probably to save money. Some functions should probably remain on dedicated mechanical switches and knobs because they're so frequently used, must be instantly accessible, or must be usable by feel when your eyes have to be elsewhere. A simple example is an audio volume control but I'm sure there are many others. But there's probably no need for dedicated switches that are used only once in a mission, and then only as part of a complete sequence (e.g., firing various pyros). Just have one big mechanical switch or button to serve as a crew "master arm" or "proceed" indication to the computer.
Lighting would be LED, of course. Not only are they much more efficient than incandescents, they're far more reliable.
All this means much lower power consumption than Apollo. So little that the thermal design would have to change, lest the cabin get as cold as it did on Apollo 13 with the power completely off.
In fact, it's an interesting question as to how little power you would really need to keep an Apollo-type spacecraft going in its cruise configuration. Most of it would probably go to the Environmental Control System for coolant pumps and oxygen fans, followed by communications depending on the required data rate (TV would take more than voice and telemetry).
