Well for starters, you typically have a limited number of vacation days a year, while sick days don’t have these limitations.
Secondly, in case of burnout (or similar), it’s not unheard of (at least over here in .nl) that you stay at home for extended periods of time, and gradually restart picking up your workload. That kind of thing is very different from vacation.
There are legal differences between sick leave and vacation, at least where I live. The former is required by state law, the latter isnt. Vacation hours can be accrued and have a cash-out value, sick leave will expire. There are a bunch of other rules as well.
You should never have to reveal the nature of any health conditions to an employer to take sick leave.