They're both brilliant.
However, I find Platinum more enjoyable, I'm not sure why....I guess I love lots of pokemon available in both games, but whilst in HeartGold/SoulSilver, they all come later on (but there's no training spots), in Platinum, there's a steady stream. I don't think there's a single route where I can't catch a pokemon I like/like it's evos. (first 3 routes, I like every single pokemon available, even Kricketot, due to Kricketune's bad-ass cry). HeartGold/SoulSilver would be inferior on their own, but Pokewalker brings it up to about level.
I think part of HeartGold/SoulSilver's problem is the amount of Kanto pokemon....whilst it'd have pathetic variety if it didn't have them (first routes still do), I've usually found them in LeafGreen, or Emerald, or Diamond, or Platinum, depending on their commonness. Both storylines are good, I like Hg/Ss for its' simplicity (although it's been given the legendary-centric treatment which screwed it slightly), and Platinum for it's general good storyline. And, tbh, if Giovanni wasn't available, and Apollo didn't have Houndour/oom, I'd completely hate Team Rocket. Galactic are just...better. And Cyrus is completely hateable, I honestly felt like murdering his sprite once, whereas Team Rocket Execs seem a bit.....unambitious, and follow like sheep trying to find their leader.Galactic survives without its' leader for a bit (until Mars gets all hormonal on Charon xD)
However, I'm just going to vote for Platinum because I'm playing that at the moment.