It's mostly down to the search being backed by Elasticsearch (so you would need to be able to install that on your server: http://www.elasticsearch.org/). As such, it does things like:
No minimum word length/stopwords for searches
Ability to "stem" to improve results (searching for "jump" would match "jumped", "jumps", "jumping")
Ability to sort results based on relevance, including recency-weighted relevance (higher chance of more recent results appearing before older ones, but weighted by how relevant the document appears to be)
Overall better performance, especially with a lot of searchable content (important for larger boards)
The relevance weighting sounds like it would help. Do you think if a resource has the same title as a search query, it will show up at the top of the results?
How much extra work is it to maintain Elasticsearch? I don't mind doing extra work upfront to install it initially, but I'm worried about additional work in the long run.
Also, I have a suggestion. Xenforo should clarify the weaknesses of not having Enhanced Search. I didn't know that Xenforo only sorted search results by date, so that makes Enhanced Search sound a lot more appealing.
Elasticsearch is really just a daemon like MySQL, so maintenance is roughly equivalent. I would say that it's mostly set it up and let it run.
In terms of the search results, I'd have to recommend trying it out here to see if it returns things the way you're hoping. There isn't content weighting, though titles are considered more important than the body.