First of all, there's a 3rd option. You can serve a dedicated mobile site on separate URLs, e.g.,
m.example.com, or you can take an adaptive approach whereby mobile specific content is delivered on the same URLs as your "desktop" site.
Which option is best for users?
From a design and architecture point of view, which is best depends a lot on what your customers or users want to do on mobile devices. The reason for this is that intent and behavior differs by device.
What do your users want?
Take a bank website for instance. On a smartphone, most people are likely to only want to complete transactions quickly and simply: check their account, apply for an overdraft, etc. They're unlikely to want to read lengthy product information or terms and conditions.
In that sort of instance, a responsive approach isn't optimal since the mobile site needs to offer content which is specially optimised for mobile.
So which is best depends on your business and your understanding of your customers.
Which is best for SEO?
Separate mobile site
The big detraction with a totally separate mobile site is the potential for SEO value to be split between your two sites, having to promote and optimise two separate sites, etc.
However, that can be mitigated by following Google's advice, which is essentially to use
<link rel="alternate" href="http://m.example.com"> on your desktop site and
<link rel="canonical" href="http://www.example.com"> on mobile.
The effect of this is to (a) indicate the relationship between the two sites and (b) indicate that the desktop site is the canonical version.
The adaptive approach sidesteps the above since the content is delivered on the same URLs. Google advice here, to avoid caching issues and let search engines know what's happening, is to employ the
Vary HTTP header, indicating that content changes based on
This has none of the issues noted above and so, from an SEO standpoint, probably presents the least number of challenges. The key consideration here is that above: does your content need to be altered too much for mobile for responsive approach to be viable?
Conclusion / TL;DR
From a technical standpoint, they're all fine for SEO if you engineer them properly. What's more important is that you understand what your mobile site needs to offer, and select an approach accordingly.