An easy way to prove the two are different is 'What is the equivalent to the endocrine system for an LLM and why doesn't it get tired like a mind?'
The logical problems that a lot of people run into are twofold:
1) At a very basic level, we modeled these architectures on brains (and named them after our brains), so at a very very basic metaphorical level, you can say 'these things are acting like our brains act.' This ignores the complexity of the brain.
2) We like to think that inconveniences of our minds (like the need for sleep) are "problems" that we can engineer away, rather than intrinsic components of the process. People won't like hearing this, but there's no formal reason why 'the need to sleep' is not a necessary criteria for 'being conscious,' because as far as we know, the only conscious beings out there also sleep. It's just human nature to assume we can engineer the 'good' parts of a biological system while avoiding the 'bad' while still essentially replicating the system, but that's usually not the case.