Pikachu probably has a lot more going for it. While they both have only one weakness, the major difference is that Pikachu can gain access to low-risk, high-reward moves like Thunderbolt, while Eevee has only Double-Edge which also deals damage to the user. There isn't really much movepool wise that Eevee has to offer either, while Pikachu learns a larger array of moves, despite the fact that most of it's just STAB.
Eevee can learn Shadow Ball, making it an awesome counter to Ghost Pokemon since it'd be immune to Ghost moves by being a Normal Type. It also can learn Bite, hence giving it some advantage against Psychic Types(albeit not like what Dark Types themselves enjoy in immunity). That said, I'd like the option of being able to EVOLVE my starter. The one thing that annoyed me greatly about Pokemon Yellow was that your rival was able to evolve his Eevee but you couldn't evolve your starting Pikachu unless you went to the Trading Corner and traded it to another game, evolved it to Raichu, and traded it back(wherein it wouldn't follow you anymore).
Many players would opt to choose Eevee if given the option because then they could Evolve it to whatever Eevee Evolution they desired(which includes keeping it as its base Normal Type form). You like Umbreon, go ahead and evolve it at night to have one. Love the new Fairy Type in Sylveon, same deal as before. While Eevee is a popular option due to the flexibility you can have with regards to Team Building, Pikachu for a Kanto starter is a really, really good one thanks to all those early route flying types. Plus by being able to learn Double Team(at level 15 in Yellow), Pikachu can defeat even Pokemon and Trainers it has a distinct disadvantage against(like Pewter Gym). Eevee can match Pikachu in defeating Pokemon it's not strong against but requires far more effort with Sand Attack(whereas Double Team can carry over to multiple Pokemon).
I should add that another reason people would choose Eevee is for breeding purposes. You can then breed an entire army of Eevee and create an army of Eevee Evolutions. Seeing as Rattata is an early route Normal Type that shares the same Egg Group, you can breed Eevee relatively early.