Large Language Models (LLMs) have been shown to be highly persuasive, but when and why they outperform humans is still an open question. We compare the persuasiveness of two LLMs (Claude 3.5 Sonnet and DeepSeek v3) against humans who had incentives to persuade, using an interactive, real-time conversational setting. We demonstrate that LLMs persuasive superiority is context-dependent: it depends on whether the persuasion attempt is truthful (towards the right answer) or deceptive (towards the wrong answer) and on the LLM model, and wanes over repeated interactions (unlike human persuasiveness). In our first large-scale experiment, humans vs LLMs (Claude 3.5 Sonnet) interacted with other humans who were completing an online quiz for a reward, attempting to persuade them toward a given (either correct or incorrect) answer. Claude was more persuasive than incentivized human persuaders both in truthful and deceptive contexts and it significantly increased accuracy if persuasion was truthful, but decreased it if persuasion was deceptive. In a follow-up experiment with Deepseek v3, we replicated the findings about accuracy but found greater LLM persuasiveness only if the persuasion was deceptive. Linguistic analyses of the persuaders texts suggest that these effects may be due to LLMs expressing higher conviction than humans.