What's really different about private and public debt?

I really want to get to the heart of what's in theory different about the public sector and the private sector.  This is a huge issue and one which will take me a long time to get to.  But I'd just like to point out one issue which economists (neo-classical and Keynesian) have when comparing the public and private sectors and that is their attitude to debt.

The first point is I want to make is to be aware of the potential presence of the fallacy of composition when considering any micro phenomenon which can be aggregated up to the macro level.  In particular reasoning about debt at the individual consumer's level or the individual company's level may or may not apply when you talk about the debt levels within the private sector as a whole.  In practice, I notice that there's often an implicit assumption that the same reasoning can be done at both levels.  Politicians too have for quite some time been happy to extend metaphors of good housekeeping and good shopkeeping at the individual level at the national level.

Whatever the truth, economists are most certainly in the practice of aggregating private sector debt up to the national level, and then making inferences about its size, particularly when compared against the baseline of the nation's gross domestic product.

With public debt, there is quite a different aggregating activity - debt issued by national and local state bodies.  So national public debt goes through a different degree of aggregation.  Again, that may or may not be important when thinking about both of these kinds of debt.  Issuance of both kinds of debt are clearly similar in some ways - they respond to the interest rate, for example; commentators like to see good debt-to-equity coverage, similar basic principles of economics determine the market of public and private debts.

But there are also differences.  Why is it, for example, that Modigliani and Miller has provided a kind of theoretical justification for increasing debt loads on companies, but there's no corresponding irrelevance theorem at the national debt level?  What would one look like?