It's not clear to me how Chalmers conceives of the "hard problem" of consciousness as separate from "the problem of other minds". He seems to wish to differentiate himself from those who see consciousness as an unsolvable mystery, and yet the problem of other minds is quite impervious to empirical examination.
The hard problem of consciousness:
Explain how and why we have qualia or phenomenal experiences - how sensations acquire characteristics, such as colors and tastes.
The problem of other minds:
Given that I can only observe the behavior of others, how can I know that others have minds?
Thr hard problem of consiousness asks how to embedd my subjectivity (first person’s view) into a scientific model (third person’s view).
The problem of other minds asks how to detect other human’s subjectivity from the third person’s viewpoint.
Both problems deal with the relation between first person’s viewpoint and third person’s viewpoint, i.e. between the subjective and the objective stance. But the problems are not identical.
You are correct in assessing that the problem of other minds and Chalmers are similar almost to the point of being different variations of the same fundamental question. In fact we can use an amusing reformulation of the two questions to see to both the similarity and elucidate the difference between them. One could restate the hard problem of consciousness as given by Chalmer using the concept of philosophical Zombies.
The hard problem becomes: (a) "Why aren't we philosophical zombies?"
While the problem of other minds becomes: (b) "How do I know that everybody else isn't a philosophical Zombie?"
The difference between the two now becomes apparent, because it is possible to give a negative answer to (b) while still giving a partially positive answer to (a): "Everybody else is a philosophical Zombie, but at least I am certain, thanks to my subjective experience, that am I not one.
From this we can see that the two problems arose in attempted refutations of different, almost completely opposed world views. The problem of other minds discusses the challenges of refuting solipsism (the principle that I am the only mind in the world), which is one step away from subjective idealism (my mind is all there is, there is no external world).
Chalmers on the other hand, formulated his hard problem of consciousness as an attempt to refute the physicalist worldview. The impossibility of describing subjective experience in terms of physical variables for him is strong evidence of dualism.
To summarize: the two problems are very closely related in that they are both hinged on the difficulty of physically explaining subjective experience. However the two problems are different in that they address completely different issues regarding the mind-body problem.