What is the difference between a fault (James 5:16) and a sin ( John 1:9a)?
For James 5:16, you are probably reading an English translation based on the King James tradition, which in turn is based on a family of manuscripts known as the Byzantine or Koine category in which the word is "paraptoma", meaning false step, fault, or transgression metaphorically. The RSV and NRSV translate "sin" here because they presuppose another family of manuscripts which has the word "hamartia" (sin) here. Both families of manuscripts have the word for "sin" in the case of John 1:9a. Essentially the meaning remains the same because, biblically, the word "sin" is understood mainly to mean transgression against God's religious statutes and moral commandments. In early Christianity, which moved away from the Temple cult and the religious customs of the Jews, "sin" was understood as a moral and spiritual transgression of unbelief and ethical carelessness. I should note that in the Eastern Greek Orthodox tradition "sin" gained an additional meaning from the Greek philosophical tradition meaning: missing the mark, i.e., for true humanity in God's eyes, which for Christians is to aim for the maturity of the stature of Christ (Ephesians).
The question is a good question to ask of a biblical society. The better Greek texts have the same word for sin (hamartia) with the same meaning in both passages so the NRSV, NIT, RSV, ASV all translate James 5.16 with "sins," meant in the same way in both passages. The KJV, which has "fault," is using a textual tradition, not reflected in the best texts, which has paraptomata, transgressions, which should nevertheless be understood in the same sense as hamaratia, "sins" (see its use, e.g., in Rom. 5.15).
There was a tradition at the time of the KJV translation, going back to Erasmus, which interpreted paraptomata as "slips." That tradition cheapens the force of James 5.16 and should be rejected.
I think you mean 1 John 1:9. The Greek word is the same in each of these texts ("hamartia"), and there isn't a significant difference in usage in them. 1 John 1:9 emphasizes God's faithfulness in forgiving sins, and James 5:16 mentions that if a sick believer has sinned it is good for him/her to confess it (urging confession among believers more generally) and so receive forgiveness (from God).