The GPT-4o you talk to through ChatGPT and the GPT-4o you access via the API are different models... but they're actually both available via the API.

https://platform.openai.com/docs/models/gpt-4o is gpt-4o in the API, also available as three date-stamped snapshots: gpt-4o-2024-11-20 and gpt-4o-2024-08-06 and gpt-4o-2024-05-13 - priced at $2.50/million input and $10.00/million output.

https://platform.openai.com/docs/models/chatgpt-4o-latest is chatgpt-4o-latest in the API. This is the model used by ChatGPT 4o, and it doesn't provide date-stamped snapshots: the model is updated on a regular basis without warning. It costs $5/million input and $15/million output.

If you use the same system prompt as ChatGPT (from one of the system prompt leaks) with that chatgpt-4o-latest alias you should theoretically get the same experience.