They're both amazing in their own way! Get whichever one you can get your hands, it's really dependent on the game you're playing. I know that in Kanto and Johto, you can only get a Dragonite. In Sinnoh, you can only get a Garchomp (before beating the Pokemon League). I haven't played in Kalos, but in Unova, when I went to face Boss Trainer Benga, I loaded up on dragon types (Garchomp, Dragonite, and Kingdra). Really, just get the one that's easier to find. You cannot go wrong with either one - and I mean it too.
Okay, if you want me to do an in-depth analysis, here it goes. Dragonite is a monster defensively and offensively, but doesn't have a whole lot of HP or speed. Garchomp has identical defense stats to Dragonite, with lower attack, special attack, and special defense, but more than makes up for this by being the fastest of all pseudo-legendary Pokemon, as well as an outstanding special ability. Sand Veil raises Garchomp's evasion during a Sandstorm, giving it insurance against supereffective moves. And not to mention that Garchomp learns Sandstorm in the Gible phase. So, even though Dragonite has Safeguard and Light Screen, plus the defense stats, don't underestimate Garchomp's defensive power. As far as movepools go, Garchomp has access to STABBED Earthquake, plus Swords Dance, Crunch, elemental Fangs, and Sandstorm. Dragonite has access to a normal Earthquake, Dragon Dance, Aqua Tail, the elemental Punches, Rain Dance, and Outrage. Both have different movepools, but they're both equally effective.
But there is one advantage Garchomp has over Dragonite, and that is how easy it is to pick up. In Kanto, you cannot get a Dratini until you're about to challenge Koga for your 5th gym badge. Dratini is a pain to find, especially with hundreds of Goldeens and Psyducks in the Safari Zone as well, and it's an even bigger pain first grinding to Level 55 to get a Dragonite, then grind again to Level 60 so that your Pokemon are at similar levels to those the Elite Four and Blue use. In Johto, it gets slightly easier because you can purchase a Dratini at the Goldenrod Game Corner if you're playing Gold and Silver. In Crystal and the remakes, you do get a Dratini that can use Extremespeed, but it is at Level 15, and since you've already gotten your 8th gym badge, you're going to have to grind a lot to get it to Level 45 (a good level to challenge the Pokemon League at). You could fish for a Dratini or Dragonair at Level 40 (and they're pretty common too) in the Dragon's Den, but I like Extremespeed because it makes up for Dragonite's low speed. Compare this to Garchomp. It is so easy to get a Gible in Diamond, Pearl, and Platinum. Before even getting your third gym badge, you could enter the Wayward Cave. Just because the cave is dark, doesn't mean you need Flash. If you go into the Secret Entrance (and make it without Flash), you can get a Gible at Level 17-20, and they occur about 20% of the time. Not to mention the TM26 Earthquake. And it'll soon evolve into a strong Gabite, who will prove incredibly useful against Maylene's Lucario and Fantina's Duskull (if you're playing Platinum). So, due to how easy it is to get, and at a point so early in the game, I'll have to go with Garchomp.