Another game that really reminds me of Telltale's approach is "Tales of Symphonia" for Gamecube. In that game, sprinkled around the overworld (or directly within the main plot) are occasional dialogue options on how to respond to people, and sometimes what path to take next. These responses ultimately affect each character's "Affection Meter" with Lloyd, and they ultimately have a role in shaping who you speak with in Flanoir (which causes a minor variation in the game from then on as to which character you're closest to; Colette, Genis, Sheena, etc.). While these responses do not change the story, they provide a different feel for the characters around you, which is a lot of what these games are about.
In TWD, it's not about branching the narrative through your choices; it's more about affecting your relationships with the other characters. And even if, at times, these result merely in changes of dialogue, that actually makes the experience a lot more significant than you'd think.
Think about it - what would any movie or TV show be without meaningful dialogue? The fact is, dialogue and character interaction play a HUGE role in telling a story. The fact that dialogue in certain scenes can be drastically different (in each of the 5 episodes) depending on the choices you've made, actually does help to tailor the story to you.
My only real hope is that Season 1's choices carry over into Season 2 (similar to the Mass Effect games), and that characters whose endings are currently left unanswered are brought to light (such as: what ultimately happened to Kenny, Molly, Omid and Christa, Vernon, Lilly (after she leaves the party; hopefully something related to her time with Lee and the group comes to light before she takes on her role in the comics), Jolene's daughter, the St. John brothers (if you chose to spare them), Hershel (before his story begins in the comics, depending on whether or not you lied to him, and whether or not you tried to save Shawn), and especially Clementine. That way, there won't be loose ends of the story just left dangling there with a lazy "You interpret what happened to these characters" just tacked onto the end of the season. Hopefully, this will be the case.
Overall, bravo to Telltale for creating an emotionally driven experience. TWD surpasses Heavy Rain in so many ways, IMO, even with Heavy Rain's branching plotline.