ESAs & SDs [or even SDits] have very-different skill-sets & public-access rights
animals used for emotional support are not required to be "trained" in any way; what behaviors an ESA can do on cue might very well be, 'Nothing.'
They also are not
task-trained - an SD is there to do a job that includes multiple tasks, which might include anything from reminding their handler to take a Rx to alerting them that someone said their Name, & the handler needs to reply to that person.
Other tasks would be fetching a phone from the charging-cradle, dialing 911 on a speed-dial button phone if the handler is unresponsive, alerting that a seizure is pending or Ur blood-sugar is too high, picking up the dozens of things that fall, roll, or are dropped in a day when U are in a wheelchair, & getting the coin-purse from the cashier after a purchase.
Most ESAs are allowed in *some* public areas;
anywhere that a pet-dog is allowed, they can go, too. The hardware store, feed-N-seed, pet supply store, small town post-office?... Sure.
Restaurants, hospital waiting-rooms or exam rooms in a clinic, hotels, a concert venue, movie theatres?...
No. Sorry, but no.
ESAs can live with their owners in any apartment building, HOA property, etc, that allows pets; if the bylaws or lease do not permit pets, odds are that an ESA cannot stay, either.
Psychiatric-support animals are in a grey area, & are often covered on a case-by-case basis. A dog paired with a vet who has PTSD may have full access, as the dog prevents paranoic attacks that might culminate in assaults on passersby; a dog paired with a civilian who has panic-attacks may be limited to pets-level access, because medication can be used to prevent the panic episodes that could occur during planned excursions [work, a Dr's appt, shopping, etc].
An agoraphobic person may, by ensuring that their dog meets every training requirement for public-access, ASK individual shop-owners to allow her ESA to enter - but if they refuse, they're within their rights - they have no obligation to permit a dog, no matter how compliant or well-behaved.
Physical disabilities are always more-readily recognized, especially if they are visible - an amputee in a wheelchair with an apparent-SD in harness will not be questioned about their dog, while a blind person with a Guide Dog may or may not be questioned, & a HoH person with a Hearing-Ear dog may or may not be questioned; an apparently able person with a dog is most-likely going to be questioned,
especially if the dog does not perform cued tasks.
If U are refused service: in stores or on transit
Getting defensive or irate never helps - if the manager or owner is adamant that they do not want dogs on their premises, SD or not, be sure there are witnesses that U can summon if U want to place a formal complaint, but LEAVE when requested.
Trespass will not help Ur complaint.

Contact the relevant city / county / state office, file a complaint with the date & time of the incident, & do it promptly.
Taxi-drivers can also
want to refuse service to passengers with SDs; legally, they cannot, as they are public conveyors. They CAN however refuse a drunk, violent, or verbally-abusive passenger - so be polite!
Even if the cabbie insults U or Ur dog, or Ur wife - or the past 3 generations of Ur ancestors! - bite yer tongue, persist politely, & phone the dispatcher to
explain [rather than to complain] that U have an SD, & need to go to X destination.
The dispatcher may send a different cab with a driver who loves dogs & is delighted to have a k9 passenger - try to go with the flow & be patient. Surely riding with someone who compliments Ur dog's well-tended coat is an improvement over being driven by someone who claims
'all dogs are filthy, they carry germs...'?
Last but not least:
Don't try to pass-off a pet as an SD, nor an emotional-support animal as an SD. It is a massive disservice to all those who desperately need their SDs to make daily living possible without 24/7 human aides, & it puts their access at risk.
For YOU, it may mean fines or even jail-time.
Frankly, i'm more concerned about the greater risk to rights of access for millions of folks with much-needed SDs, than i am about the fallout for an individual who tries to pass-off their pet or their ESA as a service-dog aiding them with a disability, a medical issue, or another covered problem.
Fake SDs who misbehave in public give all the working SDs & their handlers an undeserved black-eye; it damages their hard-won & hard-earned reputation for reliability & excellence.
- terry