In one case, you're making people enter into a private agreement with a private entity for a private good and service with their own private finances.
In the other case, you're undertaking a standard government function in taxation and having the government provide a governmental service to a citizen.
Secondly, "intrusiveness" is not the standard. Constitutional is the standard. Simply because one may be less intrusive in ones opinion is irrelevant to whether or not its constitutional. The government has a clear constitutional authority to levy taxes on individuals, and to provide some services to citizens as the government. It's far more questionable to suggest that the government has the ability to compell a private individual to spend their private resources on a private good/service from a private company. That opens up a whole different length of issues...for example, if we establish that its perfectly acceptable for the government to do such, they could for instance require that a person pays to undergo certain medical procedures or face a penalty.
It's not whether or not one is more "intrusive". It's whether or not one is within the scope of what the government is constitutionally supposed to be allowed to do.