Frankly, it varies wildly based on what the counter pick is, and how skilled the respective players are. For some baselines:
1.) Counter picking is less relevant the less skill involved between the players. It matters very, very little in bronze and silver games, but becomes critical when you hit diamond levels of play.
2.) In general, skill and experience completely trump counter-picking in league, this is because the laning phase is so dynamic and unique. You can outskill an opponent who takes a counter pick against you if your intimately familiar with your champion and his.
3.) If your playing @ gold levels or below, and are playing to climb, I would highly recommend you ditch counter picking until you hit plat and instead, focus on mastering the champions you play often. Counterpicks are sometimes lane specific, but not always - sometimes a champion can make teamfights a living hell for another champion.
As a few examples...
vs
Nocturne pretty much makes Xerath terrible. Not only does his ultimate straight up stop Xerath from bombarding ANYONE, but additionally, If Noct dives Xerath, there is VERY little he can do - Nocturne even has a shield that STOPS Xeraths only defense - the stunball. I'd call this a 'HARD COUNTER' as its extremely hard to play around. (Note: this is why you dont see much Xerath in high tier play - he has too many counters!)
vs
Ahh the classic - this is more of a soft counter, but Caitlyn has a small advantage in the early laning phase because of auto attack range and harass potential. That being said, a GOOD vayne will always crush a bad Caitlyn.
That help ? Goodluck!