Hi Marc,
The answer to this question really depends on just how much effort you want to put into the two websites and frankly what your resources are. There are many companies that successfully run different operations, Zappos and 6pm is one example.
The upsides:
1. Different target market - allows for different branding appealing to separate segments of the market
2. Different pricing strategy - allows for one site to be a 'marketplace' and the other, a full price site.
3. Different locations - One may target England, say, the other Wales & Scotland.
4. Dominate Google - Both sites may appear at the top of SERPS if enough SEO is thrown at them - therefore increasing real estate in top 10.
So there are definite upsides to having a dual site approach.
The Downsides:
1. The work involved in building and maintaining two websites. You have already said that one of them is not responsive, so I assume that is the case because you either didn't have the time or the funds to make it so.
2. Ongoing operational operations - uploading of content and rewriting for the two sites, banners, product photographs, ALTs, promotions - keeping everything separate will be a daily challenge.
3. NAPS - presumably the name & address is identical for the two sites, what about the host and IP addresses?
4. Maintaining a balance of attention to the two operations and serving the niches they are targetted at.
5. Marketing costs associated with two separate sites and brandings.
6. Marketplace links to Amazon, Ebay etc and associated costs.
7. Socials - maintaining two separate groups of Social Media accounts.
I ran an online shoe store for many years and we set up a 'sister' site which focused purely on Women's fashion. I quickly found that we lacked the resources to run two websites and ended up redirecting all of the second site links back to the first - just because of all the headaches involved.
If I were you and reading the 'non-responsive' comment, I would can that site and focus all of your attention on to the one main site:
1. 301 redirect the whole site page by page to the main website so that you preserve any backlink juice that may be pointing to it.
2. Write great original content 300+ words at brand and category level.
3. Write great original content 150+ words at product level.
4. Make sure ALL support pages are fully written, and optimised.
5. Make sure all META is optimised in terms of character length and relevance.
6. Make sure your site speed is as good as it can be.
7. All image Alts are filled in
8. Merge Socials
Frankly without going on, just make sure you cross all the Ts and dot all the i's when it comes to SEO and I am pretty sure that the combined effort of running one great site will far outweigh the schizoid way you are doing it now!
There are of course other issues, resources - do you actually want two brands? along with all the marketing costs? is that sensible from a business point of view?
I hope that helps to give you some encouragement.
Regards
Nigel
Carousel Projects