Well the Neo comes with short handles and a strap as well as the LLH version. It's made out of different material. It has a silkier feel than the classic nylon version and it doesn't fold up. I have both. I like them both. I especially like having a strap with a short handled bag.
I feel like the nylons are more casual compared to the neo. The neo is like one level higher in terms of materials used as EGBDF mentioned, and because it's monochromatic. I prefer the nylons as I know I can get most use of them without thinking of snagging the bag, and I love the many options for the sizes. For the same style (one with strap), I have the leather version (le pliage cuir) so I don't feel the need to get the neo.
I prefer the Neos. To me they seem dressier and a little more urban/edgier, while the Nylons seem more casual. The fabric on the Neos is thicker, satiny with a sheen, and they have a water-resistant lining inside. I like the sleek silver hardware on the Neos (matches my jewelry). Because the Neos overall are monochromatic, with the matching leather trim, I just find them easier to coordinate with my clothing. I don't wear brown much so the brown handles/flap on the Nylons doesn't really work for me unless I'm going super-casual.
Yes, different for all the reasons stated above. I have both, and find I like the Le Pliage classic nylon in the large, long handle tote and the Neo for the shorter handles and cross-body strap. I wish the Neo had a customizable option like the Le Pliage nylon and cuir styles do.