Polearms have one big advantage over swords when you're in a big gang of blokes: the guy behind you can stab too. This negates the "run past the poor suckers pointy end and stab him in the face as he waves a big stick around" strategy as the guy behind him can stab you in the face with his pointy end.
Also, polearms can carry things other than pointy ends. They can carry hammers, (warhammers) axes, (halberds, or axe-sticks as I prefer to think of them as) sideways-pointing pointy ends (warpicks) and who says they only need to carry one of these? Hell, a bearded axe is practically a polearm; the only thing stopping it from being a halberd is the size of the melon-slicer and the lack of an eye-poker on the end. (keep away from small children.)
There's also impact. You try getting a long blade, and try slashing it to get through a thin sheet of metal. Now try that with the full force of a long spear behind you. I bet you'd make a bigger dent (and a bigger hole in the flesh behind the metal sheet) with the force from the brain impaler than you will with ol' limb-lopper. Spears pack a hell of a lot more punch because their force is directed across a much narrower area. True, you can stab with a sword but you have to get a lot closer.
Finally, yes, you can get past a pike, if you're quick, heavily enough armoured or the pikeman was wearing a blindfold that day for a bet. Thats what the guys behind you and your side weapon is for. Pikemen carried swords as backup weapons when they could afford them. There's a reason, however, that knights considered themselves unarmed when equipped with a sword; knights went into battle generally with some kind of primary weapon; usually the lance, but also maces (a good can-opener) flails (will kill anything with one shot but you won't be able to use it again) javelins (Norman knights had a habit of chucking things at enemy formations) and, eastwards and even in a few cheeky western countries, bows and crossbows- anything but have to keep the enemy away with nothing more than a sharp piece of metal.
Swords are seen as special because thats the weapon you ''died'' with. Its the weapon you drew if there was literally nothing else; your spear had been bypassed by some cheeky bugger, your axe broken by your spoiled little sister throwing a tantrum that scared away half the enemy army, your bows and javelins long since used up on a besotted wager based on how many enemies you could pin to that row of trees stationed conveniently behind them, your lance snapped in twain at your frustration at your horse stealing the beer, and your flail wrapped around your buddys leaking noggin thanks to a drunken rampage the night before involving an argument over your midnight tryst with your allies sister. And even then, a dagger was often more use if your enemy got close enough; swords are pretty piss-poor against plate (though it was admittedly deadly against chain) and a sharp stiletto aimed at the enemies armpits (or testicles) was a frequent ending to battles fought out at close range once all the more awesome weapons were exhausted. Swords were popular secondary weapons but they were just that; secondary. Even cavalry swapped them out for lances, and even then they often drew axes or maces. The one time a sword is an advantage is when your enemy is totally unarmoured and you are at extreme close range; thats when you can cut the poor axe or spearman to pieces as you grapple.