If the brain is the only essential hardware, hypothetically speaking, then, like a computer, it will run (human body) the same way it was made. The difference is the brain is more like the memory chip and processor instead of just the processor.
If the processor works the same for everyone, then it is the memory chip that changes the info. If we do a brain transplant, the new processor (brain) would allow for the computer (human body) to run and would use the information within the brain (memory chip) for it's files/default settings.
I see how differentiation can cause differences of 2 humans growing up, but for clones or transplant customers, that differentiation would progress too small imo for major changes to take place. For that reason, I believe a brain transplant would be more like transferring someone into another body if and only if the uniqueness of what makes you different is subject to the brain and only the brain, after differentiation.