Re: Can an Epi Les Paul Sound Like a Gibson?
Yes, indeed, with mods. With the same electronics and same hardware, they'll generally sound the same.
Speaking of Epi vs. Gibson, FWIW, Gibsons need work to sound and play decently too. I haven't seen a new standard-production-line Gibson since the '90's that didn't need a new nut (or at least some improvements/mods to the stock one) and a fret dressing right off the bat. I am indeed picky about those two things, but even so, it seems to me that they just stopped fine tuning that stuff in the factory at some point around there. IME, Epis today often have better fret work than Gibsons do; at the very least, the frets are usually more crowned on Epis.
The Gibson hardware wins no prizes either, nor do the electronics, though they are often acceptable (and Epis are definitely worse on both counts).
Gibsons have nicer finishes based on materials alone, though it's not uncommon to see an Epi with nicer finish work than a Gibson. I've seen a ton of Epis with flawless, perfectly smooth finishes that aren't too thick, with nice burst transitions; and I've seen plenty of Gibsons with thick lacquer, orange peel, finish flaws, and godawfully transitioned bursts.
Gibsons often have sloppily routed cavities, with tons of router flash that someone was too lazy/rushed to clean up. Not sure I've ever seen that on an Epi, though mine have mostly been archtops, so it's hard for me to say.
As for body materials, Gibson uses more expensive/nicer wood...though even that isn't what it used to be as recently as 15 years ago. They've opted to and/or been forced to use less traditionally used Gibson woods in recent years.
My point is that if you will be extensively modding the guitar, regardless of what brand it is, there won't be a lot of tonal difference in the end. If you will be playing them closer to stock, the Gibson will be superior most of the time (and could be considered perfectly acceptable by many people after only a fret dressing and probably a nice nut).
I basically prefer Gibsons only for the theoretically nicer finishes (I highly prefer more traditional finish materials that break in more "elegantly"), theoretically nicer woods, and the "proper" headstocks. This preference is held knowing that said finish might not be that well executed, and that the wood differences probably don't really matter anyhow – so I'm gonna find good ones and bad ones out there. The main things for me are the finish and the headstock shape (yes, looks are important to me). That's why I prefer Gibsons over Epis as project guitars. Even with a crappy Gibson that's a project guitar, I can at least strip the finish easily to repaint, and will always have the nice looking head, unlike an Epi.
Bottom line, while I do indeed prefer Gibsons to Epis, especially for purposes of customization, I hold no views that with the same amount of TLC, setup, and quality aftermarket components, Epis are fundamentally inferior to Gibsons in the areas that actually matter to making music: tone and feel.