I have grew up with Pokémon as for the anime, but I never liked the story lines of it, it's also so plain, dull and boring. Seeing the same main character all over again with a electro shocker following him around wasn't really fitting me. I always disliked the Pokémon series even at the start always so.. plain boring.
As for the Digimon anime, I always liked the input they make in their enemies unlike Pokémon with the same old Team Rocket who pop in to steal Ash's Pokémons, in Digimon every Main enemy is different.
As for the 'saving the world' thing, well.. isn't that what Pokémon does as well? (Well, in the movies) Pokémon is just based on being a Pokémon Master, but that adventure never seem to end, therefore we will never know if it's ever going to be. On the other hand, Digimon always have a 'happy ending' in every Season, saving both worlds is getting more realistic and interesting by each season. I really like the design of the main characters and creature design way over Pokémon, something Pokémon never can beat Digimon on me.
As for the Games. The graphics on Digimon Games are a bit.. off. Until Digimon World 4. It wasn't really interesting gameplay for me, Pokémon on the other hand got good gameplay, but with the new upcoming Digimon game for the PSP i''ll be getting back to this, but so far. Pokémon has won on the game experience side. However, Digimon always has been interesting in their Games as I metioned the gameplay isn't that much impressed. But their stories in their games also will beat Pokémon.
So to make this short. Digimon always will beat in everything that Pokémon has, aside from Video Gaming.