The main reason why Vimeo is more professional (or at least that is the common perception) is the Pro plan that enables you to get in-depth insight and store HD quality videos with an all-round clean interface, as opposed to YouTube's layout that favours engagement and social interactions.
Good observation, Ben. Due to the fact that Google is the search engine with greatest market share and it also owns YouTube, that does put YouTube front and center. Even a video search on Google shows a far greater percentage of YouTube videos than any on any other platform.
If you are looking to get SEO value out of your videos, I'd definitely go with YouTube. It most definitely gets scraped more by Google, and is more likely to be returned in search results. I rarely see Vimeo videos returned in results (if ever actually).
IMO though, Vimeo has a slicker, more sveldt interface.
Another thing Vimeo does not have is the suggested videos for you to watch once you finish your Youtube video. Some are entirely unrelated to the current video...which could be problematic when using for business - ie. marketing.
Vimeo has the advantage, because you don't need a Google account to view a private Vimeo video. Also, I think Vimeo provides the administrator more control over how they want the video to be presented, and it defaults to showing HD. You can also password protect the video, which is not possible in Youtube. Some would argue the Vimeo embedded video player is better as well.