I'm no expert, but:
Turn 8 you have 1 Gold, 2 Silvers and 7 Coppers, but you've gone and bought 3 Whaves and 2 Walled villages (and still have your 3 estates). The other chap has got a 1 Gold, 2 Silvers, a Woodcutter (+2$ remember), a Walled Village and a Chapel, and has trashed all his coppers and estates. I would think you've overbought on actions there, and need more cash (that is working on the assumption that you're not going to go chapel - Chapel could well just simply be the best strat as Fabian said). You already have the +buys sorted, so more cash will increase your spending power more. The other guy is getting 9 per turn guaranteed from now on. You are not. I presume he'll just start nailing the platinums every turn now, while ditching his silvers.
I would say though that I think he's had some good luck with his Chapel as he's managed to trash 4 duff cards each time it's turned up, then finished off the last two when it next came round. I think having Chapelled away all the crap by turn 7 is pretty good going in my experience, but perhaps this is just normal for most people. He's carefully left himself with the six needed for the gold then platinum too.
Turn 9 Rainerd simply bags a platinum, buy you can only muster a Gold.
Turn 10 Rainerd gets a Colony (a touch early for my tastes, but again, I'm no expert), however you play all 3(!) wharves, yet still can only afford a gold.
Turn 11 A good turn for you, but should've been 2 plats really, though you fell short. Rainerd, unafraid of stalling just bags another colony
Turn 12 A total dud for you, but same for Rainerd too really.
Turn 13 Another dud for you, even though you played a Wharf, Rainerd is back on the Colonies. You have 1 Plat, 3 Golds, 2 Silvers, 7 Coppers, 4 Wharves, 3 Walled Villages, 1 Scheme, 3 Estates and... a Mandarin...? Rainerd has 1 Plat, 1 Gold, 2 Silvers, 3 Colonies, Wharf, Woodcutter, Chapel, Walled Village, Scheme. Personally I think Rainerd should've got one more Plat before going for colonies, but perhaps he was thinking "if I'm short, I'll get a Plat then", sometimes it works like that sometimes not. The Scheme is just for cycling, as is the WV, but the 3 terminals are a bit of a risk (IMO). You have lots of villages and draw, but that kind of engine can still stall hard, as you found out. Also, the Wharves can miss shuffles due to being a duration card. You have significantly more money than Rainerd, but he has a much better chance of getting it....well, perhaps not that much better now he was 3 colonies, but that's a pretty good reason.
Turn 14+ Your engine finally starts kicking in, but it's just a bit too late.
Personally I would've concentrated more on money with your strat. One Wharf - great. Leave the Walled Village, get a Silver. Getting 5 on Turn 5 is tricky of course, but then I think a Stables could've been a good choice instead of the Wharf as you still had your coppers in there. Another Silver on Turn 6, then Turn 7 would be different from what is there - 4 silvers, Wharf and Stables would get you up to Gold often. Turn 8 should've been a platinum for me in any case. 2 buys is good, but Plat is better.
Dunno, that's my thoughts. Don't worry so much about getting double colonies. Just get singles really fast, like the other guy. That's how I usually do it.
Edit - added a bit about chapel maybe just being the better strat in general.