What is the difference betwene a partial default and having debts partially written off?
"Some form of debt write-off" means usually, mutually agreed (between creditor and debtor) elimination of a portion or the whole of debt.  

"Partial default": I had the contractual obligation to pay EUR $100$ on Monday, but I paid only EUR $80$. Technically speaking, this is not always (or immediately) considered a "default", but a "credit incident" (alternatively it is characterized as "default on a payment" and not "default on the loan"). 

Then the reason why I haven't paid the remaining EUR $20$ is crucial: Do I recognize my obligation but I am unable to fulfill it because of financial inability? Or do I deny my obligation, questioning the validity of the underlying contract of the debt from which the obligation to pay the remaining EUR $20$ came?

In the first case, the issue is whether this financial inability is deemed as "temporary" or "permanent". If it is deemed "permanent" (usually by a court decision), then in some countries the debt is officially written off by necessity, while in other countries the creditors retain the right to seek payment whenever the debtor acquires assets in the future (so the creditors may be monitoring you). If it is deemed "temporary", then the issue is to make an attempt to restructure the debt.  For Public Debt, the inability is never officially considered "permanent" so the issue is always to restructure (which of course may include a mutually agreed partial write-off, which essentially means a recognition that there exists a "partial" permanent inability to pay).

We could call the second case, where we question/deny the validity of our obligation, a "unilateral debt write-off", which of course is rarely if ever accepted by the creditors, and so whether it should be called a "write-off" is questionable in turn.