Situation: OWD for both Account and Case set to Private. I create an Account, then click New Task. No problem.
Next, I create a Case, click New Task: insufficient priviliges.
What am I missing here?
I can think of several possibilities why this is happening.
If cases are handled by another group of Users, you may have an assignment rule that kicks in and transfers ownership upon creation. Since OWD is private, although you created the Case, you'd no longer have the privilege to create tasks related to it since you'd no longer be the Owner (just as would be the case with Opportunities related to your Account).
Its possible that your Org on provides Case support when there's an SLA of some kind that exists for the Account or only when the Account has purchased a product. If that's the situation, a new Case for such an Account, while it might get created, it might also not go any further when the SLA isn't found.
If you no longer own the Case due to an Assignment Rule having fired, your ability to access or do anything related to the Case could, such as creating a Task, would depend on Case Sharing Rules and your CaseTeamRole, assuming you're a CaseTeamMember.
I'm certain there are other possibilities, but these are the 3 that I could think of.
This genuinely sounds like an issue related to Sharing and Roles related to Cases and Assignment Rules. I don't know if new Accounts go through an Approval Process in your Org, and if so, whether that would affect Cases or not. There are a myriad of possibilities.
I suggest you check your work rules, assignment rules, sharing rules, Case Team Membership, Case Team Templates and Case Team Role assignment rules if the latter three are used along with any Service Agreements or Entitlements related to Case support in your Org that you may have in place.