I have many colleagues that fly both the 300 and 800. But from a technical level, to add. The NG has a larger wing. Slightly more chord but an extra (un de-iced) slat and simpler flaps that seal together rather than separated by the pylon. The -300 is really a re engined 100/200 wing, with deeper leading edges to up the chord and the triple slotted flaps and slats are 727 items. The engine pylon is a kind of adapter from the -200s underneath engines to the CFM hanging off the front. With the NG it has a cleaner more traditional pylon.
The problems IMHO opinion of the NG are that the Aileron system was unchanged, so you've effectively got semi inboard ailerons, that simply do not have the power to hold against full rudder and compromise cross wind performance as a result. They are CRAP.
Flightdeck, overhead is more or less the same. The -700 uses a -300 type air con panel but the -800 uses the -400 system with both packs operating to cool to the coldest of the three settings and trim air used to warm the other two sectors. I've flown the -700, and the -800 and the trim air system works far better.
EFIS Screens are obviously different, there are more covers to the window frames and a drip tray at the top of the windscreens to stop condensation dripping down. Better Electric gauge panel up top is the other improvement.
It's still a1967 arrangement of 1955 parts up front mostly, just as uncomfortable and noisy. But I'd take it over an airbus for the feel and awareness of what's going on.