You need a doctor's prescription to live in no-pets housing and/or travel via plane with an ESA, but if you living in housing where pets are allowed and don't intend to travel via plane anytime soon a prescription isn't really necessary, because ESAs are basically just prescribed pets. ESAs only need basic obedience training and, with the exception of planes and no-pets housing, they aren't allowed anywhere regular pets can't go.

Depending on your individual situation a service dog may be more useful, because they're trained to preform actual tasks for their Disabled handlers and are able to accompany their handlers in public. If it helps, here's a list of the type of tasks I'm planning to train my Autism service dog: deep pressure therapy (DPT), mobility/balance assistance, light guidework, hearing alert, and non-aggressive guarding (ie: block/cover). I'm also considering training him to alert to tics and retrieve/carry items when my hands are ticcing, but that's for a comorbid, undiagnosed disability and thus won't be considered an official task.

Other important info: If you live in the United States there's no need to register either service dogs or ESAs. Every registry you'll find online is a scam, and there's no legal value to the "certification"/"registration" papers they provide. If you're living in no-pets housing you need to submit a reasonable accommodation request along with a doctor's note prescribing either an ESA or a service dog.