As others have mentioned, you avoid "payroll taxes" (Medicaid, Social Security, etc) by using pre-tax money rather than post-tax money.
However, there is one benefit to getting your own privately held one: you can choose the service provider. A previous employer's HSA charged $4/month, and did not allow me to invest in any funds unless I had over $4k in my account. However, a single year's maximum contribution is less than $4k, so it was stuck in a money market account perpetually. The
The tax saving probably is larger than both your monthly fees and your investment gains, but the HSA provider's rules are another (fairly-opaque) consideration.