Traditional or trad refers to games that harken back to the original RPGs, like Dungeons and Dragons. They tend to be more mechanically-based and more complicated. Indie games, on the other hand, follow a more narrative tradition and major examples include Fiasco or Apocalypse World. They tend to be mechanically lighter and more driven by the tropes of storytelling.
Here's the thing. A lot of the ways that I have heard people break down the two categories tends to be more about the philosophy of the people playing them as opposed to the actual design itself. You can have a character driven, story based game using a traditional system. And you can have a conflict driven Game that's all about fighting with an indie system. Some of them, it won't work very well but you can still do it.
If you strip away the people playing the games and just look at how the systems work, here's what I think you will find the real difference between the two is. Traditional games are about creating a concrete world for you to interact with. Indie games are about creating a story for you to interact with.
Hopefully, you're going to get a story and a concrete world in either case but I think each philosophy has it's own priority.
Another way that I personally tend to think of them is that traditional games are about confrontation and indie games are about conversation.
Traditional games come out of wargaming. As a general rule of thumb, if most of the rule book is about fighting, you're probably looking at a traditional game. In fact, Shannon Appoline wrote in his book Dungeons and Designers that in the earliest days of role-playing, the game master actually was trying to win against the players.
While traditional role playing games have developed a lot more depth compared to the days when it was just about beating each other up, it is still build around interacting with the environment. That environment can include obvious things like weather and rain or more esoteric things like monsters or politics. And there are mechanical ways of resolving all of your interactions with them. The rules have built a world for you to interact with and that is what drives the system.
On the other hand, indie games are driven by the story. The actual world itself takes a backseat to the needs of the story. The game is not about how you do something but how you tell the story.
For me, the game that really made me realize the differences between the two philosophies was Polaris. Polaris was not the first indie game that I ever paid but it was the first one that made me fundamentally realize what an indie game was trying to do.
You see, at its heart, the rules for Polaris resemble a great we simplified Roberts Rules of Order. They form the procedure for the players to debate with each other. I had never seen anything like it and I still think it is pretty amazing to this day.
Both philosophies are perfectly valid and both will give you an engaging experience when done well. Traditional games will give you a world to be drawn into and indie games will give you a story to be drawn into.
At the moment, I am much more into indie games. Part of that is because they take less time. I don't have to worry about the mechanics I just need to worry about story and that something that I understand innately. But I also have a long history with traditional games. They are also the deep and enriching experience.
These are two different voices that are important to the hobby.