> Cursor seems kind of "dumb" compared to 4o

What do you mean by that? You can use 4o model in Cursor?

I think part of the reason is, ChatGPT has additional system prompt in chat, API doesn't have.

even if both companies use same model (OpenAI and Cursor), system prompt has a huge impact on results

Cursor tries to save more money on context than ChatGPT. So when both are targeting GPT 4o, the ChatGPT App will incorporate more context history and memories than Cursor.

Cursor lists a large number of models out of the box, and you can select which one you want it to use. https://docs.cursor.com/settings/models