The wording of the two statements is certainly a little ambiguous, but I think we can find at least one plausible reconciliation. Only JKR knows exactly how to resolve this; consider this more a strong conjecture on my part.
Let’s go through the statements, and I’ll explain what I think.
When a Secret-Keeper dies, their secret dies with them, or, to put it another way, the status of their secret will remain as it was at the moment of their death. Everybody in whom they confided will continue to know the hidden information, but nobody else.
The phrase “their secret dies with them” is used by Muggles, and I think we can draw some useful parallels. If a Muggle has a secret, and they never tell anybody or write it down before their death, then it literally dies with them: nobody else will ever know what it is.
But suppose they told three friends first, and asks them not to tell anybody. (In other words, the original Muggle wants to retain control of who has the secret.) Nobody spontaneously gains knowledge of the secret upon their death, and their friends don’t all forget, but now their friends who are responsible for deciding who to share the secret with.
Magic merely reinforces the trust of friends. If a Secret Keeper tells you their secret, then you are bound by magic not to tell anybody. (This is why Harry had to learn of Grimmauld Place via a handwritten note from Dumbledore, and not another member of the Order.)
If you were standing in front of Grimmauld Place when Dumbledore was killed, then it wouldn’t suddenly appear in front of you. Conversely, if you were there, you wouldn’t suddenly forget about it. (“Everybody in whom they confided will continue to know the hidden information”) But with Dumbledore’s death, the dynamics of the secret have changed, and responsibility for keeping or confiding the secret rests with those who Dumbledore told.
However, if Dumbledore had never told anybody, then nobody would ever be able to find it again. (A more practical application for this might be some sort of treasure or prize.) That’s how I interpret “their secret dies with them”.
It’s a bit of a loose interpretation of JKR’s statement and that phrase, but I believe it is just about consistent.
Mr Weasley had explained that after the death of Dumbledore, their Secret Keeper, each of the people to whom Dumbledore had confided Grimmauld Place’s location had become a Secret Keeper in turn. “And as there are around twenty of us, that greatly dilutes the power of the Fidelius Charm. Twenty times as many opportunities for the Death Eaters to get the secret out of somebody. We can’t expect it to hold much longer.”
Let’s interpret this: Dumbledore had a secret. He told twenty people (members of the Order). Dumbledore dies, so the responsibility of keeping the secret transfers to the Order members. Just as they would have the responsibility of deciding whether to share a Muggle secret, so they have the responsibility of keeping Dumbledore’s secret. The rest follows.
I’ll recap briefly:
If the Secret-Keeper dies without telling anybody the secret, then nobody else can ever discover it. If they tell anybody else, then responsibility for keeping the secret is transferred to those people upon their death.
Other explanations
There are two other plausible explanations:
JKR is just inconsistent. (Dismissed as being out-of-universe and not very interesting.)
The Fidelius charm used by Dumbledore to protect Grimmauld Place is different to the usual charm.
Let’s unpack this second possibility. Suppose we take JKR’s statement to be an absolute one: with the standard charm, when the Secret-Keeper dies, the only people who know and can ever know are those who were told by the Secret-Keeper before this death.
Dumbledore is smart, so he probably foresaw his death (be it that of Half-Blood Prince or not), and took appropriate measures. The Order needs to carry on after he dies, and that means they need to be able to bring new people to Grimmauld Place. Dumbledore may thus have set up the charm so that Secret-Keepers work as I described above, so that this could happen.
I think this is another plausible reconciliation.
Notes
I’ve not written enough, so I have another spanner to throw in the works: can we reconcile this with the Potters’ home? Peter Pettigrew, the Secret-Keeper, doesn’t die for almost seventeen years after the attack. But it’s clear that magical entities have been visiting their home and leaving tributes for a very long time. There’s probably a subtlety here about whether the secret is itself alive (similar to horcruxes), but I don’t have time to discuss it now.
