I think it's a mental thing. Seth is still going to be under someone else' shadow. Seth knows he will still have to perform at his best, but he won't have to think about the media comparing his stat numbers to his brother's stats for the same games. On a different team, they are playing with different players and against different teams so you can't put down direct comparisons.
Going to team practices and being told to haul in the water bottles would be embarrassing.
Going to pre-game media interviews and being compared to your brother's stats would be embarrassing.
Going to team Christmas parties and being told to haul in the drinks and napkins while the players' wives set up the tables.... and the rest of the players are partying... he's a legit second-class citizen.