What is the difference between "oof" and "oops"?

Can I interpret that "oops" is for when you yourself make a mistake and "oof" is for when someone else has a slip up? Do they share same origin? They seem awfully symmetric.

The Free Dictionary defines oops as:

Used to express acknowledgment of a minor accident, blunder, or mistake.

For example:

Oops! I forgot my library card at home.

Merriam Webster defines oof as:

used to express discomfort, surprise, or dismay

For example:

Oof! She just dislocated her shoulder, is there a doctor?

"Oops" always indicates an error (literal) slip, or clumsiness.

"Oof" does sometimes share this meaning, but it's onomatopoeia for the sudden expulsion of air when punched in the stomach. It's more an expression of unpleasant surprise than error. Wiktionary has this use first, and M-W doesn't quite define it that way but the example does the definition I give. Many other dictionaries only seem to have it as a slang term for money