Stuck is frustrating. Stuck is scary. Stuck still has hope of breaking free. Stuck usually means something is wrong with the story somewhere (and that's fixable.)
Burnout is paralyzed. Burnout is numb. Burnout says 'I can't'. Burnout usually means something is wrong in your environment or possibly in your physiology.
Burnout needs recovery time and effort and may require professional assistance either from a coach, a therapist, or a doctor. Stuck needs a little strategic psychology to pull you free of the mire and can usually be resolved within a few weeks. Burnout can take months or even years.
Whether you're in burnout or whether you're stuck, the antidote for the poison is the same: So long as you say 'I can't', you won't. The only way out of stuck and out of burnout (eventually) is to begin asking 'how can I?' Example: Going from 'I can't make this work.' to 'How can I make this work?' It's not magic and it's not immediate. You have to ask the question and ask the question and ask the question. Then you have to sit and listen. Free write around it with no rules. Stream of consciousness write around 'how could I make writing work?' 'What would it look like to make writing work again?' 'What do I need in order to write again?' Just keep asking and keep noodling. Initially, the answers will all be I don't know. Then, one day, you'll get an outlandish, crazy idea. Mine was to get some plywood up into the attic and laying it down across the rafters so I could get a desk and plug up there as well as a little damned peace and quiet and solitude for writing. Living in Florida where attics are regularly a bajillion degrees made the idea unusable but it did start a protracted effort to find some privacy in an overcrowded house. It's s process that's still unfolding. Recovery is a process that takes a little self awareness, honesty, and a willingness to ask for help if it's needed.