In-form American Spieth, fresh from winning the Australian Open, was five under through 17 holes in his second round when play was suspended due to darkness at the Isleworth Golf and Country Club.
Spieth is at 11 under overall, two shots clear of Swede Henrik Stenson (68) " who is one ahead of American Patrick Reed (63) and Englishman Justin Rose (64).
Woods, who is the tournament host and five-time champion, did manage to improve on his dreadful first-round 77.
The 14-time major champion carded a two-under 70 to get back into three over, but he is still last in the 18-man field.
Spieth would be in an even better position if not for a double-bogey six at the par-four sixth, but he lifted on the back nine.
The 21-year-old holed five birdies on the back nine, including three in a row from the 12th.
Stenson made back-to-back bogeys at holes 15 and 16, while Reed eagled the par-five seventh.
Rose was inconsistent " making just one par on the front nine as he mixed six birdies with a bogey and a double bogey.
His back nine was far better as he produced five birdies to be in contention.
Woods improved on his poor performance, showing flashes of his best which included an eagle on the 13th.
Despite being on track to improve his position in the standings after 17 holes, Woods faltered on the 18th after a rain delay, double-bogeying the last hole after two poor chip shots.
"I struck them all solid yesterday and I did again today," Woods said in a press conference. "But obviously I hit it a little closer today and made a couple putts."
Americans Bubba Watson (68), Jimmy Walker (69) and Rickie Fowler (70) are tied for fifth at seven under, a shot ahead of countrymen Keegan Bradley (66), Chris Kirk (68) and Zach Johnson (71), as well as Australian Jason Day (67).