Beginnen wir mit den Vorteilen.

It covers medical expenses 
A private health insurance policy covers certain medical expenses that public health system or Medicare doesn't.

It covers (some) hospital expenses
In the event that you'll require hospitalization, a private insurance policy could cover some of your expenses, and you'll be treated like a private patient.

And now for the cons - and there's a big one:

The cost
This will be a real problem if you're already in poor health and don't have any kind of group coverage. Usually, individual health insurance policies can cost around a few 100s of dollars a month.