When you infect a computer and use a 0-day exploit, evidence of how you got in is often left behind. Preventing yourself from leaving any evidence is about as hard as having software that has no exploits in it; next to impossible.
Many computer systems aren't patched regularly; on such a system, an old exploit will usually get you in just fine. This exploit being discovered ... doesn't do much. I mean, if you took over 20% of the computers on the internet with a specific exploit, you might notice an increase in patch rates. But you might not.
A 0-day exploit, on the other hand, can be used to break into security-conscious targets. If you care about the specific target, and they work at being secure, the 0-day exploit may still get you in.
Your attack may, however, be noticed. And once noticed, they might work out your exploit. And once they work out your exploit, they could share it with the vendor, who might patch it; or they might hack a patch themselves.
And now, your 0-day exploit has patches published, and every security-conscious system on the planet blocks its use. So tomorrow, when you really want to break into a secure server somewhere, you'll need different and new exploit. You burned your exploit.
Not every use of your exploit is going to be noticed, and not every notice is going to result in a patch, but every use increases the chance that a patch will arrive that breaks your exploit.
We can illustrate this with some examples of state-sponsored computer hacking. Stuxnet used four zero-day flaws (that there was no security against). Its discovery led to all 4 being patched, "burning" their usefulness in the future. In exchange, a pile of expensive centrifuges in Iran broke, slowing Iranian nuclear research.
It did the job of multiple cruise missiles, with far less diplomatic, humanitarian and military risks.