In this case — the VPS going to COO at only $250k in ARR — well the COO is doing nothing. It will be a failure. No SaaS company needs a COO at $20k in MRR. You haven’t figure out how to scale. Do not hire a COO until you have figured out how to scale. Period.
But at a higher level, as you scale, a great COO can make all the difference. As you hit $10m ARR (or earlier, if you are growing super-quickly), a COO can take over everything the CEO isn’t core to.
Once you cross $5m in ARR, at the latest, as CEO you shouldn’t be VP of anything anymore. A COO can help own some of the functions. If the COO is great at revenue, perhaps, she can own Sales, Marketing and Customer Success (although having sales report to the CEO is almost always better). Or sometimes, the COO can own Product and Engineering and Support. Or sometimes, literally, operations.
What a COO can’t own is fundraising, PR, closing most VPs (COO can recruit, but CEO needs to close), and vision.
All the rest is open for grabs.