To have your dog as your ESA, you need to visit a doctor/psychiatrist and get a proper note after the doctor speaks with you and decides you have a need for an ESA. ESA's can only be allowed in cabin on flights and in non-pet housing. That's the only special privileges they get.
A service dog must perform a task. They also go through training and it sometimes takes two years before a dog is ready to be a service dog. A service dog is trained to abide by public access standards when it accompanies its handler in public. It performs tasks that help mitigate its owner's disability.
An emotional support animal is more of a dog that provides comfort, and doesn't generally do any trained task for disabilities.