When Hawking argued that information is lost in black holes, this triggered the "black hole war" because it threatened determinism, which would mean the laws of physics are only true on average. But quantum collapse was also considered to lose information, but this was not seen as such a threat. Can anyone explain why these two issues are different?
Quantum collapse absolutely does threaten determinism. In fact, it not only threatens it but smashes it in the usual sense of the word. There is no amount of information I can gather (even asymptotically) so that I can predict the precise output of a single instance of a general quantum experiment.
This is something that theorists have mulled over so much that they stop mentioning it at some point. But it is no less true.
What the information loss paradox is concerned with is a weak, quantum form of determinism. This determinism can be, strictly speaking, only formulated within quantum mechanics. It states that if we are able to repeat an experiment an infinite number of times and thus to reconstruct the quantum-mechanical wavefunction corresponding to its output, then we can deduce the evolution of this wavefunction to be deterministic. Technically, this is tied to the unitarity of the evolution of the quantum state.
The information paradox could mean that even the evolution of the wavefunction itself will not be deterministic. This is a violation of determinism on a different level then the quantum-mechanical collapse.
However, I cannot personally say that I understand the amount of attention the information loss paradox receives. This is because the operational definitions of wavefunctions and quantum states are inherently non-cosmological. Once we start asking about global, cosmological questions, we cannot assume that, for instance, we can let a universe evolve an infinite amount of times to reconstruct its wavefunction. Quantum theory is by construction a local, lab theory and it comes - at least to me personally - as no surprise that it runs into difficulties once applied outside of that context.
Quantum collapse case:
Quantum collapse is just a phenomena of collapsing a set of quantum information to one of the quantum information upon measurement. By repeating the measurement we can achieve the whole set of quantum information. And thus not to consider as a deterministic threat(more accurately: "quantum information is never lost"). Also, looking through many world interpretation gives the meaning that the set is never collapses to one, but exist all along the way in different realities.
Black hole case:
Black hole evaporation via Hawking radiation leads to information paradox because after the evaporation, there is no way to recollect the lost information. But there may exist solution(s) to this paradox like Black hole never evaporate completely and thus information can never lost.
Also, for the comment of @Charles: No cloning theorem says that you cannot clone a quantum state in anyways or in other words, quantum information can't be copied. This doesn't tell you about wavefunction Collapse or lost quantum information.