In most cases, paying a cash price will not be less expensive than going through your insurance. But, it never hurts to check.
For instance, a standard office visit may be $100 (just using easy numbers here). For a cash-paying person, they would owe $75. If you have typical health insurance, you would generally pay a $25 or so copay and insurance would cover 100% of the cost (aside from the copay). Some insurance plans may have you pay a $25 copay and they will cover 80% of the office visit, making your total cost $45. Some insurance plans pay nothing until you've met your deductible. So, it just depends...but I'd say in most cases you're better off going through your insurance.
I'm doing infertility treatment and my insurance covers zero, so everything is out of pocket for me. The clinic I previously went to did give a small discount for certain things (ultrasounds and IUIs) but now I'm going to an actual infertility clinic which is more expensive. It's $325 just for an office visit.