Bad light and rain prevented play from going ahead in the third session after Mitchell Owen (70no) helped Tasmania recover to 6-166 by close of play at the SCG.
Neil-Smith (7-58) picked up where he left off following a rain-affected first day, dispatching of veteran quick Jackson Bird to break a 52-run partnership with unbeaten Ollie Davies (81).
The Taswegians made light work of the Blues from there and when he bowled Chris Tremain for a duck, Neil-Smith had career-best bowling figures with the hosts bowled out for 224.
Neil-Smith's figures were the sixth-best in a first-class match for Tasmania since the turn of the century and the best since fellow paceman Sam Rainbird took 8-21 two summers ago.
But just as their NSW counterparts did on day one, the Tasmanian batting order misfired early.
Bird (3-46) took the first three wickets as the visitors fell to 3-30 on a tricky deck, having survived a dropped catch and appeal on their way to 0-29.
Things threatened to spiral further out of control when Moises Henriques, enjoying a great summer with the ball, had rival captain Jordan Silk caught behind in the over before lunch.
In at eight, Mitchell Owen inspired a fightback, partnering with Jake Doran (42 not out) in an unbeaten 96-run stand that helped the visitors bounce back from 6-70.
When the covers came on, 22-year-old Owen (70) was four runs shy of equalling his best first-class score.
While fielding, Davies suffered a hairline fracture in his left little finger but the Blues are hopeful he will return to bat during their second innings.
Davies, NSW's saviour on day one, looked on track for a second first-class century but was left stranded amid Neil-Smith's dominance early in the day.
