"My initial reaction is that sway
has a neutral or slightly positive connotation (the lawyer tried to persuade the jury); while influence
has a neutral or slightly negative connotation, and suggests a range of possibilities from persuasion to unethical conduct. But my initial perception could easily be swayed or influenced with more context."
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The lawyer tried to sway the jury = O advogado tentou persuadir (talvez comover) o júri. Aqui o advogado lançou mão de recursos positivos.
The lawyer tried to influence the jury = O advogado tentou influenciar (talvez com algum estratagema, talvez com alguma inverdade) o júri.