Sorry Bluesy but there just ain't no contest.
No, they do NOT use the same woods. The Gibson woods are of higher quality and the construction is far better. It is for this reason alone that model for model the sound is different. Yes, I agree that for some models it gets darn close but it is not the same.
I'll give you an example from personal experience:
Many, many years ago I had a Gibson ES345 1964. Like so many of us on here I sold it and have regretted it ever since. I din't ever think I would own a similar Gibson again.
When Epiphone brought out their version of the ES345 I was elated. I ordered one in advance and made extensive high grade modifications. It was a damn good guitar no question.
But then I came by a bit of money unexpectedly at the same time Gibson started making reissue ES345s. I could not resist and got a 1959 reissue ES345. Not the same as my 1964 but - in my opinion - even better.
I was obviously able to directly compare the Epiphone with the Gibson. The difference was astonishing. Just no comparison. The Gibson was leagues better. The Epiphone was nowhere near the same class (and sound).
I got rid of the Epiphone.