I agree with Joseph. I would add that an argument is compelling if it is so strong that it FORCES (compels) you to agree, and leaves no other reasonable choice. In contrast, an argument can be convincing, and yet not be so strong that you feel like there is no choice but to agree with it. That said, in actual usage, people will usually treat the two adjectives as synonyms in this context.

If you used this in real life, "compelling" sounds fancy and would sound weird in some situations. I work at as an engineer and would never say "that is a compelling argument" ドラマみたい. I would say "that is a convincing argument". Also, compelled and convinced are very different words. For example, "I am convinced that it will rain tomorrow", but "I am compelled in will rain tomorrow" does not make sense.

Good question! The difference in this case is very subtle. They are indeed very close synonyms and both are equally correct and mean virtually the same thing in this context. I would say that "compelling" is a little stronger of a word and if an argument was "compelling" that you were REALLY convinced. Think of what it means to be compelled vs convinced. If something is "compelling" to you, not only was it convincing, but it also affected you emotionally whereas "convincing" has no emotional connotations.