Is it the same to say 'related to' instead 'connected to' here?
It especially centers on the life of lead singer Freddie Mercury, who died in 1991 from health problems connected to AIDS.
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Yes, "related to" sounds fine here. In this context it is the same as "connected to."
Of course in other contexts the two words might not be interchangeable at all. "Connected to" literally has the meaning of "physically attached to", and "related to", literally, is about kinship. Here, the phrases would be used in a figurative sense, and either works.