I play a 1998 LP Classic. The distinction with the Classic is that it uses
a 500T bridge pickup and
a 496R neck pickup. These pickups are higher output than the standard pickups (or at least they were in 1998 before the BurstBuckers became standard), lending a bit more growl and bite from the stock guitar. Also when I bought my LP Classic, they were the only series of LP that had the 60's slim-taper neck.
So the classic is the same instrument as a LP standard, but with different pickups, the slim-taper neck, top-hat volume and tone knobs, and some faux-aged appointments like the greenish fretboard inlays and tuning keys. Oh, and the cream-colored pickguard.
It still has the mahogany back, maple top, neck and body binding, and Tune-O-Matic tailpiece that
the LP standard has.
When I played my Classic side-by-side with a comparable LP Standard and a '76 Custom, I thought it had more of that classic LP "bite" when you really dig in and play hard. I also thought that the slim-taper neck was superior to the clubby-feeling standard LP neck. That always bugged me about Les Pauls, that the neck was always so awkard feeling. But the slim-taper neck is much more comfortable for me. I have large hands so its not a matter of better access or reach...its just that the radius and width is much more comfy for me.
So there's a first-hand scoop on at least on 1 more LP model for ya!