To be honest, there's a number of ways to go about it.
Running two websites would be an option, but that'd be a lot more admin in the long run. It can be good for trying to get more of a market share though, two results in the SERPs for keywords, as opposed to one.
From an SEO perspective, so long as Google likes its backlink profile, it'd possibly make sense to join the domains together. You could do this URL by URL using a 301 redirect.
So for example, if they had a page which was related to barbeques, 301 redirect their barbeque page to your barbeque page. And so on. This will combine your backlinks and should, in theory, help the root domain a lot.
Saying that it doesn't mean it's always the best option. I'd get somebody who really knows their stuff to analyze the new domain.