It depends on which “Model” of Les Paul.
The sustain is great because it’s a set neck (rather than a bolt on neck). Folks have preferences (I prefer set - but I’ve loved a few bolt on neck guitars as well). For example, if you break a set/through neck (or the head), you’re kinda boned - can be repaired but probably wont sound/feel the same way. With a bolt on guitar, just unbolt and bolt on a new piece.
The wood of the body really colors the sound too. I LOVE mahogany bodied guitars. It’s just a “darker”/“deeper” sound. (the Les Paul has a Mahogany body - with Maple top). I’ve had Ash/Alder bodied guitars (both with single coil pickups as well as humbuckers) and they just sound too bright for what I like.
For Mahagony wood tone, think of the great Gibson guitar players - Angus Young of AC/DC, Jimmy Paige, Slash, etc. For Ash/Alder, think of the Fender players - Jimi Hendrix, SRV, Ritchie Blackmore, etc. It’s not to say one is more suited than the other for various types of music, it’s just preference.
Beyond that, pick ups and amp dictate the sound (both are very much replaceable - less so than body/neck type).
Then there’s the play-ability aspect of it. The Les Paul can be pig heavy. There’s actually three body types. One is the “Solid” which would be crazy heavy - but have the best tone. Then there’s the “Traditional Weight Relieved” that’s commonly known as the cheese holed ones (there’s 9 holes routed into the body). My LP is Traditional weight relieved. Then there’s the modern routed one which is essentially like a semi-hollow with a solid maple top (which a lot of traditionalists scoff at). But it’s an effort to keep things light.
Their Custom Shop Les Paul get’s first dibs on wood choices. They’re typically solid bodied, but are light weight (lighter than the traditional weight relieved ones as well as the routed ones). But you pay for the guitars. 
I went with Traditional Weight relieved (9 cheese holes) because I wanted the heft for sustain/tone. I was actually fortunate too because my guitar was less than 10lbs. Typically, they’re heavier.
The nice thing with Gibson guitars is that they have a Nitro Cellulose finish vs a Poly finish that pretty much everyone else uses. It’s thinner and smoother to feel and let’s the wood breathe. But again, preference. One thing I don’t like about the Les Paul is that it’s hard to play (for me at least) when you get high up in the frets (15-17 onwards). lol that’s when you start doing the thing Slash does by playing the guitar vertically.
http://wac.450f.edgecastcdn.net/80450F/loudwire.com/files/2011/09/slash1.jpg
Check out some of the Ibanez Prestige guitars. They’re made in Japan. Specifically, check out the RGA (RG shape with a contoured from) or the “S” series. They’re both Mahogany bodies (regular RG isn’t). Very comfortable body. Neck is super thin (if you don’t like the baseball bat contours). Sounds great (stock pick ups are junk - kinda a joke with Ibanez, swap em out for something else). Great price for what you get.
I had a REALLY nice RGA Prestige that I sold to fund my Les Paul mods. I regret selling that guitar so much…
https://lh6.googleusercontent.com/-VN7ohDQie2E/S7khfov396I/AAAAAAAAAWE/7_E1v7ZHO-o/w1152-h768-no/IMG_7528.JPG
I swapped out the pickups for some good Seymour Duncans (I had the one that Van Halen used to use in the bridge, and a really nice sounding one for the neck).
