I have one of these, though I am not a Les Paul expert by any means.
By way of comparison, I have owned a LP Studio, Standard, Silverburst and a chambered R9 (I think it was). Except for this current Axcess, I've not really gotten along with any of the LPs I've owned in the past.
The Axcess is quite a bit lighter (its not "chambered" though, weight relieved is what its called I think) than most others I've played, obviously the heel is very different and really comfy and the back is carved as well. The body also might actually by a little bit thinner than your "normal" Les Paul. The neck is more like a 60s carve, so thinner than most. My guitar had the Burstbucker in the bridge, but I got rid of that for a Suhr Aldrich which I like very much.
I think the con is also the pro...the Floyd Rose. Some guys think its an abomination, but I like it quite a bit and the tuning stability is really nice. To be honest I've not played another LP since I got it, so I can't say if the lighter body has much effect on the sustain or tone of the guitar, but with the more powerful pickup there is definitely no issue...I didn't like it so much with the stock bridge pickup.
Does Gibson even make these anymore, or are all now Alex Lifeson models with the Piezo and two outputs?