The developers of Saints Row took the idea of the Grand Theft Auto Games and took it in a whole different direction. They didn't stick purely with what they knew would work, they took risks. The Saints Row games started with a more serious tone than you'd expect if you were introduced to the series with SR3 or SR4. GTA stuck to the same type of story throughout all of their games (although it was impressive) and made the safe move of playing it safe and sticking to what they knew would sell. Saints Row innovated and riskily changed the direction of the game series and included new things such as comedy and sci-fi elements. The Saints Row games have some of the best lines in any game while GTA keeps their dialogue only mildly entertaining. GTA became big purely because of the shock value of the first few games. Saint Row pokes fun at every aspect of GTA at every opportunity possible. As for the gameplay both games are near the point of perfection but there are many more options of things to do in Saints Row after you finish the story. Honestly I don't think I can name a single activity in GTA that you can't do in SR. GTA is the mainstream choice and I understand why but it seems like Saints Row simply has much more to do than GTA. If you want to know what game is "Better" then you need to know what you're looking for in the game since they're both entirely different, entertaining experiences.
Grand theft auto has been around since 1998, so this means they have had more time to develop than saints row. Gta III obviously led to saints row, not to mention that it revolutionized gaming forever. Grand theft auto is one of the most critically acclaimed series of all time, and it's games are much more polished and sort of down to earth than saints row. Saints row is still awesome, but it's wasaaaaaaay to over the top (4 at least) and it's much more buggy and glitchy and hasn't progressed nearly as much as the gta series has. For gods sake, they used the same exact city in half of the games. That's lazy.