Their economies are radically different otherwise.
Greece has a weak economy in most fields, with the exception of tourism. Japan is a manufacturing and scientific powerhouse.
Greece runs recurring high deficits and had rarely, if ever, shown inclination to stop doing so. While Japan was criticized at the start of their financial decline for insisting on balanced budgets instead of stimulating demand.
Tax collection. Japan does it. Greece didn't.
Japan could devaluate its currency if necessary to adjust its finances. Greece couldn't, being on the euro.
Japanese government debt is, I believe, mostly owed to Japanese investors and denominated in yen, unlike Greece's.
Japan has demonstrated for decades that it can pay its bills, on its own. Greece on the other hand just seemed as if the only thing that kept it up was euro membership.
Greek official statistics were not, and had not for a while, been trusted. They're still prosecuting the guy who showed they were cooking the books.
Until 2008, Greece was only getting charged a 0.25% point premium over German bonds based on the tacit assumption that Europe (Euro-zone) would never let Greece go broke. Investors, out of greed and risk un-awareness (or trust in European taxpayers gullibility), parked their money in Greek bonds, rather than German ones, just to get that extra 0.25% point.
From https://www.bankofgreece.gr/Publications/Annrep1999.pdf (I find it hard to set hard time limits on Google to only look at stuff from before a date):
Specifically, the yield differential between the Greek and the German 10-year
bond fell from roughly 270 basis points at end-1998 to about 200 basis points in March
1999 (see Chart VI.1).
At the start of the general crisis, investors started wondering if trusting that Europe would always back Greek debt, even when undergoing a Europe-wide crisis, was a wise idea. Once perception soured, the game was up very quickly. Financing costs for Greek government debt went up, fast. Any refinancing happened at progressively higher rates. A debt that looked bad at the previous, lower, rate seemed like it would result in a short term default. People still loaned to Greece, but the premium to do so was massive and made this very quickly unsustainable.
The plug was pulled, late high-rate private and non-European investors got a haircut. Europe's taxpayers paid up enough money for Greece to keep paying slightly lowered interests, at deferred times, to the big German and French banks so they wouldn't take too massive a loss (and so that loss of confidence wouldn't spread, as per @Machavity answer).
Most Greek debt remained in place, unlike what happens in say an Argentinian default. In fact, money "helpfully" loaned to keep Greece from defaulting on interest payments just got added to the overall debt. In a way, though it fully deserved the initial mess it got, that's the tragedy of Greece. Instead of the reset that comes with a default, Greeks got this unending misery of servicing a debt which is only pretended to be fully recoverable from (as most of my post is critical, I want to stress that, IMHO, the Greek people don't deserve to live through this for decades).
But no one is too willing to put their hands in that wolf's jaws anymore.
As o.m. says, mostly perception. Until COVID-19 at least, Japan looks like it has a few good years left for investors, most of whom are Japanese and have little interest in rocking the boat. This is not to say it is financially healthy, only that it is under less external pressure.
Greece relies on foreign money, didn't look good in 2008 and doesn't now. In most other conditions, big chunks of that 181% would have been written off as unrecoverable already.
