I've got accounts at both.
I started banking with a credit union in High School. I kept a couple of savings accounts with them that I'm still depositing into.
After I got my career going I got several checking and investment accounts with a large bank.
Large banks have more leverage on the economy which they can use to benefit themselves and accounts they manage. Credit Unions are owned by those who have accounts with the union rather than being publically traded and thus take less financial risk and have better savings rates. It's also easier to access your money anywhere in the planet when you are with a large bank then it is when you are with a credit union. But big banks have more account fees then Credit unions.
I would say have both to get the benfits of both.