What is the difference betwene a partial default and having debts partially written off? From The Daily Telegraph:


  The new Syriza-led government has demanded some form debt write-off on the country's €317bn liabilities, two-thirds of which are owned to official creditors in the form of the EU and the IMF. 


From BBC News:


  New Greek PM Alexis Tsipras says his country will not default on its debts.


A default is, according to Wikipedia:


  In finance, default is failure to meet the legal obligations (or conditions) of a loan, for example when a home buyer fails to make a mortgage payment, or when a corporation or government fails to pay a bond which has reached maturity.


Which leads me to the question:

What is the difference between some form of debt write-off (demanded by the Greek government) and a partial default (denied by the Greek government)?
"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.