For a technical comparison of the Classic Vs the NG have a look at this website:
The Boeing 737 Technical Site
From a Pilots perspective the NG is a huge improvement on the classics. The APU is far more capable, it is designed to run both packs so it's always warm in the winter and cool in the summer. It also provides far quicker starts than the APUs fitted to the classics. The Flight Deck is now fitted with 6 large screens (DUs), the PFD and ND now sit side by side, and the engine and systems DUs are in the middle one above the other. The lower DU, which displays secondary engine information and some system information is customised so you will see variations in displays here. The overhead panel looks much the same, except for perhaps the pressurisation panel which is now a simpler digital system, which you may see on very late model classics as well. Other systems appear the same but there are some huge improvements in the "behind the scenes" architecture, particularly the electrics.
Flap 25 is now a take-off option on the NG, although I have never used it and I don't know anyone who has. The climb is not as "sporty" as the classic, and climb speeds are generally faster than the classic, but the rate of climb is less, (assuming similar CIs being used). The NG is certified to 41,000', whereas the classic is limited to 37,000'. The NG is faster, again assuming similar CIs the NG cruises at c.785/460kts, whereas the classic is c.745/440kts. The fuel flows are about the same, but with the higher speeds, the NG is more efficient.
The NG appears less speed stable and is prone to overspeeding in the descent if you are not careful. Boeing supposedly addressed this problem revising the PATH profile back in 2005 but it is still something to be wary of. Likewise the NG is not as speed stable on the approach either IMO. The -800/900 series have a very high approach speed, which I believe to be driven by the need to keep a lower body angle and avoid tailstrikes, rather than being a function of the stall speed. This problem is addressed with the short field performance kit.
The NG's autopilot is now cleared down to 50' (single autopilot operation), and the NG is also CAT IIIB capable (I believe). For navigation the NGs have GPS receivers, and a number of regulatory authorities will allow RNAV arrivals as well as non-precision approaches to be flown using the autopilot and LNAV & VNAV. The NGs can also be fitted with a Head Up Display. Other nice touches on the NG are the Flight Path Vector, and a metric altimeter.
IMO the NG is the nicer to fly, the thrust pitch couple is almost non-existent, but it's perhaps a little less predictable to land.
The NGs FMC benefits from a bigger memory and a faster processor over the classic's, otherwise they are generally similar but there are a number of customer options, meaning you will see differences from one operator to another.
The NG being a bit faster is also a bit noisier. Newer models now have a row of vortex generators ahead of the windshield, which is supposed to drop the noise level by 3Db IIRC.
I am led to believe that the NG was developed on a minimal budget, so whereas it is a big improvement on the classic it could have been so much better. I believe the wing is a scaled down 777 wing, but doesn't deliver 777 speeds. The flight deck should have been upgraded, but a major customer (think Southwest) dictated that it should retain commonality with -200's let alone -300's. (there may well have been certification issues too). I would also like to have seen some of the QRH checklists improved, I am thinking here of some of the flap checklists, and the one engine inop. landing checklists, with Vref (ice) and the go-around procedures. These could/should be fixed by either a design change, software change or wording change.
Overall a good reliable workhorse, and I hope that's given you some insight into the aeroplane.