With only about an 1.5 hour of time with the guitar, after it arrived today, I have to say I am reasonably pleased with the differences. To answer some of my own questions, and echo comments of some those who contributed here, the LP Custom is very different from my 594s and Tremontis. I plugged into my normal rig and did not adjust anything so there will be additional changes in the future. This is simply a direct comparison rather than saying where either could work better.
I found the LP Custom to be more mid-focused than the Tremonti and the pick-ups were somewhere between the 594 and the Tremonti in terms of output. The volume pot is far less versatile than any of the PRS Guitars. I can use the full range on my PRS, while the LP bridge pick-up drops off tremendously after ~7. Overall fit and finish is very good, but with far more noticeable "handmade" features. The nature of hand-made binding leads to far more surface intersections and slightly noticeable changes between the neck/body and binding. Frets are great (all of the guitars are PLEK'ed at the factory) and I am impressed with the feel of the fingerboard which is smooth (ebony). The 498T is much clearer than I was expecting and should work well.
In closing, I see a lot of similarities between my PRS's and the LP Custom (as it is to be expected), but none of them seem to be direct replacements for any of them. The LP Custom feels really good to play and will fill a spot in my fleet.... though it is not going to take a top-3 spot away from my SC594, Tremonti, SC594 soapy. It will be in a comfortable 4th place tie with my DC594 and still getting plenty of attention.
Since this is a PRS focused forum, I will not do a full NGD review/discussion on the guitar, but I figured if I included a PRS in the picture I could at least close out my original questions. Thank you all for the feedback and suggestions.
As an interesting (to me side note), I was talking with my wife about a PS build and her only comment was "Please do not make it another grey guitar"... and that was before this one showed up
I guess I have a "type".
Little late to this thread, but I went through a similar process about 6 months ago when I was looking to get a Les Paul Custom Reissue 57 and doing a bunch of research. Les Paul Customs really vary by year, and even within the same year or family of build (reissues, Norlin era, etc.). Its insane how many variations there are amongst Les Pauls and its why its recommended you always play the one you want before buying, unless you have a return option. PRS' on the other hand, are pretty consistent from year to year. Obviously there are some exceptions, but you're never going to get two PRS in the same family of guitar that are wildly different. The biggest differences between the LP Custom and 594 are...
--- Les Paul's are much heavier. My Custom Reissue I think is like 9.5 lbs, the LP Standard is around 9 lbs, and my SC594 is bout 8.5 lbs. I used to have a doublecut 594 and that was even lighter, around 8 - 8.3 lbs. Most Customs you won't find under 9 lbs. The norm for most LP Customs though is around 9.5-10 lbs, with the occasional 10.5 lbsers in there.
--- All have different bridges. SC594's bridge is bolted on, so it won't detach if you remove all the strings and it also feels a little lighter/lower to the body. A LP could have an ABR or Nashville bridge depending on what you get. The big difference is with a historic reissue Les Paul Custom. My 57 reissue comes stock with the taller pickup rings which I personally don't like. I replaced mine with the shorter standard LP rings, but even still, the bridge is raised up much higher than it would be on an LP Standard or SC594. A "standard" LP Custom should have the regular pickup rings and the ABR bridge vs. the Nashville bridge used on a reissue (57 or 68).
--- Pickups are all different. I think LP Customs from the mid 00s to around 2017 used to come with Burstbucker 1 and 2s (Alnico 2). The new ones now come with 498t bridge and 490R neck, which are Alnico 5 and 2, potted, and much hotter pickups (8k vs. 14k ohms). 594's come with the 58/15 LT pickups (Alnico 2). I think those are around 8K ohms. The 58/15LTs are nothing like either of those pickups and feel much closer to the Custombuckers in the Les Paul Reissue guitars IMO. I actually prefer the 498T for more hard rock/metal stuff, and replaced the Custombuckers in my Reissue with the 498T, and about to do the same in my LP Standard by replacing the Burstbucker with a 498T. But if you play more blues based, lower gain stuff, the 58/15LTs or Custombuckers are nice pickups. I like the 58/15 LTs second after the 498T because they feel like they can actually play more harder rock/metal type stuff if that's what you're into, but also the cleaner stuff. They are somewhat bassy though. The Burstbuckers are good for stuff in the classic rock area, but they aren't as good for heavier stuff. Custom 24s have the 85/15s, which are closer sounding to the 498T/490R, but with more bass/low mids. Its a darker pickup, but sounds good under gain. I've never tried the Tremonti's, but have the PRS Metal pickups and imagine they are close to that (both around 16k ohms). Much hotter, but have really good cleans and sound clear/defined under heavy distortion/gain. Nothing like the 498T which is a much looser pickup like the Duncan JB.
--- The biggest differences and what matters most are the neck profiles and whether or not you like or can handle the neck/body carve. The "standard" LP Custom comes with a C neck around .83 at the first fret I believe. From the ones I've tried, I remember it was pretty similar C carve to the 594, except the 594 felt a little more comfortable. Again, this can wildly vary by LP though. Even the DC594 I used to have felt like it had a slightly smaller neck compared to my SC594. LP Reissues though, come with giant C shaped necks. To give you an idea, my LP Custom Reissue 57's neck is about .91 at the 1st fret. The LP Standard 50s (more like a D shaped neck) is about .87 at the first fret. The SC594, a slim C shaped neck, is like .83 at the first fret. I think most "standard" LP Customs C shaped are around the .83 mark as well, but from the few others I've played, they varied wildly too. I think the Tremonti and Custom 24 necks are the slimmest at around like .78 or .80.
TLDR: My advice to anyone, if you're looking to buy a LP Custom or 594, try them out and see which one has the better neck profile, neck joint, or overall weight that you prefer. It doesn't matter how the guitar sounds unplugged or what pickups are in them. Pickups are an easy swap and you're not going to play an electric unplugged anyways. What you can't change is the neck profile, neck joint, or overall weight of the guitar. The neck joints on Les Paul Customs suck. They're bulky and blocky. The 594 will be a much more comfortable playing guitar past the 12th fret, more modern feeling, stay in tune better, and be much lighter while standing up. What neck profile or size you like though, is all personal preference. If you like chunkier necks (like I do), then you'll like LP Customs. And nothing quite has the sound, "grunt", or thickness like a Les Paul.
I personally like both guitars for what they are. Here are some sound clips of my 594 and LP Custom. Even playing through different gear, different pickups, or direct vs. my cell phone, they still sound pretty similar. The LP Custom has the 498T in the bridge and 594 has a Holy Diver, which are pretty similar pickups. The LP Standard with Burstbuckers (which older Customs used to come with) and PRS Custom 24 with Metal pickups on the other hand, sounds pretty different from either the 594/LP Custom.