What's the difference between and ESA and a PSD?

An emotional support animal (ESA) is a pet that helps its owner by providing companionship, helping relieve stress and otherwise comfort its owner just by being present. ESA owners’ rights to keep an ESA in a rented home are protected under the federal Fair Housing Act (FHA). The law requires landlords to allow ESAs even if they have a No Pets rule or limit the types or sizes of pets renters can keep in their homes.
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You’ll need a licensed medical professional to diagnose you with a qualifying condition and provide you with a valid ESA Letter to protect those rights. Unlike psychiatric service dogs (PSDs) and other service animals, airlines and businesses are not required to allow you to keep your ESA with you. 
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ESAs do not have to receive intensive training but should be well-behaved. That is important because, while qualified ESA owners have specific rights when it comes to living with an ESA, bad behavior like constant barking, aggressiveness, or destructive actions are reasons that a landlord or property manager can have your pet removed or make you move out.
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A psychiatric service dog (PSD) is a dog trained to perform specific actions to assist someone to deal with the effects and symptoms of a recognized psychiatric disability. To provide that help, PSDs need obedience and situational behavior training. But they also need extensive, individualized training to provide the complicated help individual users require. We discuss those tasks and training requirements elsewhere in this FAQ.While ESA owners’ rights are limited to housing under the Fair Housing Act, the use of psychiatric service dogs, like other service animals, is also covered by the Americans with Disabilities Act (ADA) and the Air Carrier Access Act (ACAA). Those laws protect PSD users from discrimination while traveling on airlines and other commercial travel providers, in housing, and in access to most private businesses and public spaces. As long as their PSD is properly trained and can be reasonably accommodated, Psychiatric Service Dog users are entitled to have their PSD living with them in their home; in an airline's main cabin; and when they go to appointments, public events, and go shopping, out to eat, or on errands to other businesses.