RedRover.org outlines the important differences and requirements on this topic:

Emotional Support Animals (ESA) and Service Animals are covered under the Fair Housing Act, so they should pose no barrier to finding housing. Your client must have an ESA letter issued by a qualified doctor or therapist for their pet to legally qualify as an ESA. These letters are typically good for one year. 

Keep in mind that ESA do not have the same protections as Service Animals under the Americans with Disabilities Act (ADA). Only a fully trained animal is considered a Service Animal. Service Animals in training are not protected by the ADA. 

If your client is disabled, they can hire a trainer, or train their own dog, to get their dog to perform a task/tasks to assist them as a Service Animal (only dogs and miniature horses qualify as Service Animals at this time). Here is a resource listing some of the Service Dog tasks for psychiatric disabilities (panic disorder, PTSD, depression, etc.)

Here is an FAQ from the US Department of Justice on Service Animals and the ADA. 

In situations where it is not obvious that the dog is a service animal, staff may ask only two specific questions: 

1. Is the dog a service animal required because of a disability? 
2. What work or task has the dog been trained to perform? (Staff are not allowed to request any documentation for the dog, require that the dog demonstrate its task, or inquire about the nature of the person’s disability.) 

There is no national registry for Service Animals or ESA. Your clients do not have to carry a special card, have their dog wear a vest, or pay a registration fee to have a Service Animal or ESA. If they desire a vest (which can help people to maintain their distance), both Service Animal and ESA vests are available through online retailers like Amazon. There are even wallet cards available that can help people explain their rights regarding their service animal.