The Gibson Custom Shop is where they make their elite line of guitars. A lot more care goes into wood selection. A lot more of the work is done by hand. It's here that they make copies of their original guitar lines from the '50s and '60s. Even though many of these models still exist in the normal, factory line-up, they've been changed, usually for cost-cutting reasons.
For example, a modern Les Paul Standard, is mostly hollow, because the mahogany they use for the back is of a kind that would make the guitars too heavy if it were solid. It has some effect on the tone, but some people prefer it, others don't. But the VOS (vintage, original specification) models made in the custom shop, use only lightweight mahogany. These guitars therefore can have entirely solid backs without being too heavy. They will thus tend to sound more like the original classics. Other changes involve using cheaper electronics in some of the newer guitars, different hardware, etc..
Here's a link to a recent picture of Reinhard's beautiful Custom Shop 1959 reissue Les Paul:
http://www.guitarforum.co.za/your-gear/flame-on/
There are some changes in the body shapes - for example the Gibson SG today looks a little bit different from those in the early '60s. Also, a few of the factory guitars, the Gibson SG Standard, for example, have two piece bodies (Les Paul Classic too I believe), while the originals - and the VOS reissues - are made to the same specs as the earlier versions.
There are also quality issues with Gibson's factory production. I get into fights with a lot of people who feel it devalues all Gibson guitars to say this, but the evidence is overwhelming. On the first page of the mylespaul forum it is highly unlikely that you won't find a new thread on any given day asking for help because their guitar came with serious flaws. This doesn't mean they're all flawed, just that the quality control isn't what it should be for such highly priced instruments. The majority are great guitars, but there are a significant number of poor setups, serious issues, or outright duds.
The Custom Shop, on the other hand, has a great reputation for quality. The problem with them usually, is that they're crazy expensive - especially new. The suggested retail price for their copy of a 1959 Les Paul Standard is about $9,000. That's why I favour Japanese clones like Tokai, Burny and Edwards - that make VOS, or near-VOS clones of these guitars sometimes for less than even the Gibson factory models, and with fantastic reputations for quality.
Here's a link to my Tokai:
http://www.guitarforum.co.za/your-gear/tokai-ls150-new-pic/
But with the weak dollar, the Custom Shop guitars are getting pretty reasonable second hand. As I said, a copy of their 1958 Les Paul can now sometimes be had for nearer to R15,000 than to R18,000 on ebay.