Basically, service dogs are specially trained animals. Trainers have worked with them extensively to respond to some aspect of their disability, like with a seeing-eye dog that helps a blind person navigate their community. Or a dog that has been trained to respond to seizures in someone who has epilepsy.
Emotional support animals are much looser. They typically aren't specially trained. My experience with emotional support animals is that someone got one after their psychiatrist or therapist essentially prescribed the need for one. I've seen people with emotional support dogs, cats, birds, and rabbits. But emotional support animals really seem to tool to help people regulate their depression or anxiety -- not through their any special training, but through ongoing relationship and proximity.
It's worth nothing that the Americans with Disabilities Act (ADA) covers and protects service dogs and allows them access to public establishments. The ADA does not offer the same level of protections or access to emotional support animals. In other words, a seeing-eye dog should have full access to a restaurant or hotel. An emotional support dog wouldn't have full access to a restaurant or hotel.
Incidentally, service dogs might have full rights and access to public establishments. However, they can be barred for being aggressive or out of control. Additionally, they need to comply with all of the local laws regarding licensing and vaccinations.