I *think* the difference came down to this - your opponent had more money.
Here were a few points where he got that extra money.
1) Your opponent bought Fishing Village as his second Village, where you got a vanilla Village.
2) Once, early on, he used a Spice Merchant for +$2 to buy a Conspirator. You had a situation where you used it for +2 cards, and later in that turn bought a 2-cost crossroads. (I think that was shuffle luck, since that turn you drew 2 copper, 2 estate, 1 spice Merchant. But still, he had a point where he used it for an early $2 for a conspirator).
3) The turn after that, he used 6 to get Conspirator+Crossroads.
4) And then, YOU used 6 to buy a Council Room (which does not give you any cash!)
By this point in the game, he has four Conspirators to your one, and he had enough villages to activate them. Your Council Room even helped him activate them - at least one of those turns, he didn't have the cards to start a conspirator chain until your CR gave him an extra one. He had enough in his deck to buy provinces now, whereas you still had to take two more turns to buy some more conspirators - and that was the difference.
The game was still pretty close; the final score was 39-31, and even though the province split looked like it was 6-2, it could have easily been 5-3. It was a difference of 1-2 turns or so. A single shuffle going the other way could have swung it.
That's my interpretation of when he actually got ahead of you. Really close game though, I can't claim that I would have done any better.