Some do, some don't.
Luckily I've had a pretty understanding employer. That is... until I completely failed them. And am in the process of leaving my job out of my own imaginary shame.
Here in Estonia most people think of mental illness as:
1. Made up or a sign of weakness (depression, anxiety)
2. Incurable and uncontrollable (schizophrenia and psychotic illnesses)
People just cannot imagine mental illness and there is nothing to educate them except for appalling, inconsistent and incomplete news articles. We're not taught in schools about such things early on.
These are some of the reasons I came up with why people/employers treat mental illness from physical illness/injury different.