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.
Oops! I forgot my library card at home.
Merriam Webster defines oof as:
used to express discomfort, surprise, or dismay
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
Based on both your question and your repeated comments on the various answers already present, it seems like you're wondering about the trend of young people today using the word "oof" in very casual online conversation. (Though I've heard it used verbally as well.) From what I can gather, it's a very informal word, used jokingly/sarcastically. It seems that's the general consensus from Urban Dictionary as well:
when you don't really care but should say atleast something
As a native English speaker, I have not seen or heard the word "oof" used anywhere in any professional setting.
"Oof" as a reaction tends to be reserved for mistakes which carry some significant consequence for the person making the mistake - someone embarrassing themselves socially or accidentally injuring themselves. "Oops" tends to lack this connotation of pain or distress, and so tends to be used more broadly, say for a minor misstep or gaffe.