So I've got my ES-335, my Sheraton, and my ES-339. 

My Gibson ES-335 is a 2016 Studio version.  As I recall, it has a few differences from the regular ES-335 model, but, overall, it's still the same guitar.  It has the 4-knob arrangement (unlike the Studio version from a couple of years earlier), and it has 57 Classic pups in it.  I think the bridge used was a different bridge from the regular version.  I also think that the neck is a torrefied neck.  I think there were a couple of other minor differences, but those two are the most significant changes in the Studio and regular versions.

The Sheraton is a 1962 50th Anniversary model.  It's biggest differences between it and an actual 335 are that it uses Gibson mini-buckers instead of full size ones and it has a Frequensator tailpiece (closer to a trapeze) instead of using a stop piece tailpiece.  It has CTS pots and switches, and it has GIbson cloth wiring inside.  Those are also differences between it and the regular Sheratons of the time that it was built.  For the purpose of comparison, when I bought the Sheraton nearly 10 years ago, it cost approximately 50% more than a regular line Sheraton and half the price of an ES-335 (which is the model I would've gotten).

My Epiphone ES-339 is from the second run that they produced.  The first run was during the summer or fall of 2011.  I had been looking at and toying with the idea of buying a Gibson ES-339 for a while when they announced that Epi would be producing that model as well.  I got my order in late enough that I missed that first run and had to wait on the second.  The only difference in the two that I recall is that the first run had Epi's ProBucker pups in it (their equivalent to Gibby's BusrtBuckers) and the second run had the Alnico Pro pups in it (their equivalent to the 57 Classics).

Although, mine have some pretty significant differences in them, the 335 and Sheraton are very similar guitars in theory.  However, I believe that the Sheraton was an Epiphone creation and not a Gibson copy originally, but it's a little more ornate than the basic 335; it's closer to an ES-355.  I actually like my Sheraton better than my ES-335, and, moving forward, I'm going to talk about them as if they were the same unless specified otherwise.  Also, I'm not covering the differences in Gibson vs Epiphone.  This is 335 vs 339.

The 335 guitars are bigger bodied.  I'm too lazy to look up the actual dimensions, but they're huge guitars.  They're big enough that I know at least a couple of folks that won't play them because they're "just too big."  They're also pretty heavy.  I attribute the weight to the fact that they're so big.  Although I've always said that the 339 is the same size as a Les Paul, I've read (and seen when they're side by side) that they're not quite the same size.  But they're close.  See the pics at the bottom for a comparison.  It's not a huge difference, but that little bit makes a pretty big difference in weight and comfort when playing.

If it makes a difference to anyone, the 335 has the jack on the face of the guitar where the 339 has it on the hip like a Les Paul.  I don't usually think about where the jack is, but every time I pull out the 335 and plug it in, I'm afraid I'm going to hit the plug and crack the face of the guitar or break the jack.  I've never done that and probably never will, but it's always in the back of my mind.

From a parts standpoint, they're not really that different of guitars (and I'm not talking about the the Gibson vs Epi difference, this is the 335 vs 339 difference).  They both have the same control setup.  Both have 2 vol/2 tone setups with the pup selector down by the knobs.  They both have a stop tail bridge.  They both have humbuckers for pups.  There's the jack location, but, other than that, they're similar.

Tones from both are very nice.  They both have that nice semi-hollow sound.  The longer I play, the more I really like that airy sound that you get from a semi-hollow like a 335 or 339.  The difference is that the 339 can get into those Les Paul-ish kind of darker sounds that a 335 can't do.  I imagine that's because the wings of the 339 are smaller so it's as close to a solid body as it is to a semi-hollow.  The 335 is the standard for a semi-hollow, in my opinion, the semi by which all others are measured.  The 339 does an adequate job in that semi-hollow arena, but it also can get into the LP territory.  The 339 is a good balance between the two. 

Once I understood that what I was hearing was a semi-hollow sound, I've liked the 335 and played one at least as often as I played my Strats.  I've always wanted to like a good Les Paul, but have just never bonded with one.  Since I started to migrate from primarily playing an acoustic to playing an electric, I have always been drawn to my Strats.  However, about a year ago, I pulled my 339 out of the closet, set it up really well, and have been playing it almost exclusively since then.  It's just a great guitar that covers a lot of area.

The 335 vs the 339.  Both are great guitars that are very similar.  If you like the humbucker sound, give them a shot.

And, yes, I've ripped those from the interwebs at some point.  Don't remember where so I can't give credit like I should.