For quite a long time I have been using these two words interchangeably without thinking of their difference, even if it is slight. So can someone differentiate the former from the latter, please?
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In "discrepancy" the stress is, IMO, on showing a "discordant" difference.
Related to DISCREPANT
Synonyms: clashing, conflicting, disagreeing, discordant, inconsistent, incompatible, incongruous, inconsonant, inharmonious, mutually exclusive, repugnant
Merriam-Webster Unabridged Dictionary
I should be startled to see "discrepancy" used in any context other than some distressing mismatch, e.g. in accounting or between somebody's story and other evidence of the reality. Of course you could use "difference" there as well, but the substitution the other way does not generally work. I should never, never use "discrepancy" for many other differences, e.g. "there is a discrepancy between British and American English".