It'd seem to be a pure matter of nuanced semantics, but it's quite a fundamental difference. it's best to understand it as we understand insurance in a more traditional scenario: for cars. 

We don't want to get into accidents. And more often than not, we are rather attached to personal, fashionable belongings such as our cars—we name them; we spend a bunch of time in them; etc. Point is, we try to avoid using the car insurance that protects us from financial ruin when we [knock on wood] get into a car accident. 

We don't want to use our car insurance. We DO want to use our health insurance.

Same is the case for health insurance—somewhat. It's supposed to work like this. It's there to protect us from financial ruin in case we get gravely ill. We're not supposed to want to exercise on those benefits, and yet, soon as we get us some good ol' company sponsored health insurance: we're like kids in a candy store, and the first thing we try to do is max out our benefits: seeing all kinds of doctors, seeing if we can get whatever tests done that we can get ordered. 

But that's not the real intent and purpose of health insurance; it's supposed to be there to protect us from financial ruin in case we get gravely ill. So, there are these professionals in the health insurance industry called actuaries and statisticians and economists and financial engineers—whose job it is to come up with the physics and math that make up these financial products, these health insurance products, for us to use and subscribe to. The big objective of these professionals is to balance the rate at which benefits are exhausted against the rate at which policy benefits are not exercised. 

That's not the case with a plan. A health PLAN is built to be used; is built from the ground up to be exhausted. That's why government sponsored healthcare often comes in the form a health PLAN, not health insurance—but a PLAN. These are services that members are expected to utilize. 

It's like a cell phone plan or any other kind of plan— a family plan on Spotify for instance; you're EXPECTED to use that service.

That's the fundamental difference between INSURANCE and a PLAN. With insurance, it's like the approach to healthcare is one of balancing odds and a matter of risk management. With a plan, the economists that drive that whole—well, "plan"—are doing so fully expecting participants in the plan to exhaust all of the plans wonderful features and uses. 

Some common features of health insurance policies: 

  • Patients or beneficiaries are referred to as subscribers, i.e. "Blue Cross subscribers."
  • Financial responsibilities are of the kind / type / scenario of: co-pay, co-insurance, deductible, etc. 
  • Types of health insurance products include HMO, POS, PPO, EPO, etc.
  • Big insurance companies include, Aetna, Blue Cross, Blue Shield, Cigna, Health Net, United Healthcare, etc.

Some common features of health plans: 

  • Patients are referred to as members, e.g. "Health Plan of San Mateo members."
  • Often, financial responsibilities of members are referred to as "shares of cost"— if ever the member has one. 
  • The only reason why a member would even have a share of cost on a government health plan would be because the patient / member didn't "totally qualify" for the plan to begin with (didn't meet low income brackets). 
  • "Shares of cost" are listed in eligibility verification results for, for example, Medi-Cal (Medicaid) members.