It is well known that both hardware and software contributed to the total dominance of computers. Has there been an experiment to run a very modern program (say Alpha Zero) on a museum computer (of course, a memory issue can happen, so don't abuse the Zuse for your amuse) or vice versa, a museum program on a very fast computer? If yes, with what results (hard- or software more important)?
The answer is software is more important. You can test this yourself actually. Get your favorite top engine today and your favorite top engine from 10 years ago, and run both of them against each other at time odds. This simulates superior hardware (since it yields a higher number of positions searched).
This was done at 50-1 NPS (nodes per second, a measure of how fast the hardware is) odds here. Komodo 8 (a top 2014 engine), played on a smartphone against Shredder 10 (top 2006 engine) on an i7 desktop. Komodo won 5-1. Not a large sample size by any means, but clear enough given auxiliary data.
See also How does engine strength scale with hardware?, which in conjunction with rating lists played on equal hardware gives an idea of where hardware effects might dominate over software.
I develop mobile chess apps to the App Store. The very first version of the apps were built on the iPhone 3GS. That was many years ago, but even then the apps could play at the GM chess level.
There is a rating list for Android chess engines:
You can then compare the list with more powerful hardware configurations:
No rating list exists for iOS due to technical limitations.
According to the JCER rating list, there is a big gap. Stockfish on Android is
3155 while the PC version is
Note that there is a massive jump for the software side with the introduction of neural networks like Alpha Zero. Software before that required a large amount of clever humans programming it but these humans could do that with low end hardware. Neural networks need in comparison much less human programming time but require lots of computation power once to train them. After the training is done they can be run on worse hardware than pre neural network software and still play much better.
This makes numerically comparing pre neural network to post neural network software difficult because it is not clear what exactly you want to compare. One could run a modern chess software on say year 2000 hardware and it would trash any software available back then but I don't know how feasible it would have been to train a modern chess software with year 2000 hardware. A chess software probably can't afford to rent a super computer for a month of time for training.
"Turochamp", the original chess-playing programme from 1948 by Alan Turing and David Champernowne, was executed manually for a game in 1952, since hardware at the time was deemed incapable of running the software proper. It took about 30 minutes per move, and lost to Alick Glennie after 29 moves.
In 2012, for the Turing anniversary celebrations, the programme was reimplemented on contemporary hardware, losing to Garry Kasparov after 16 moves. Kasparov explained that the basic problem was that Turochamp is "only two ply", that is, it cannot look ahead very far and in particular could be deceived by a sufficiently skilled player who was aware of this limitation, regardless of the speed at which it would produce its moves.
As a caveat, the 2012 programme was not the same as the original, as the authors didn't preserve all details of the algorithm, and may not have implemented it thoroughly for the 1952 game in any case. Nonetheless, this demonstrates a long time gap, and that even with much, much better hardware, the algorithm is still not that capable.
I don't see how any experiment would decide the question of "importance".
If you tried to run modern software on ancient hardware, it would fail, because modern software is designed to take advantage of modern hardware.
If you ran ancient software on modern hardware, it would do whatever it did before, only faster. Would it beat modern software running on modern hardware? Almost certainly not, because it wasn't written to take advantage of the hardware power available.
But consider: a lot of the improvement in computer chess is due to the existence of databases of openings and endgames. There's no clever software behind that, just availability of cheap storage. So do you attribute that to software or hardware advances?