Ammon Johns: `Better` is a rather vague term, since sometimes cheaper is better, and sometimes more advanced (which costs more) is better, etc. On that alone, Alan pretty much nailed it. No one solution is `better` in every case, or every regard.
Where performance is the primary criteria, to the extent that the client is prepared to spend more in both time and cash to get it, I still recommend a dedicated mobile site. This and a dedicated app are still the only way of providing an experience that is absolutely and fundamentally built to give a great experience on mobile without having to compromise.
Neil Cheesman: Perhaps to evaluate how best to serve the website visitors - and in terms of workload, will there be extra work in `managing` two sites and could that extra work be better spent on improving/marketing one site?
Alan Bleiweiss: All answers so far come up short. There is no single answer for all situations.
Everything depends on which method your developer(s) are willing/capable of implementing with the highest level of confidence regarding accuracy of implementation.
I typically recommend responsive because it does mean developing and then maintaining only one code set, you maintain full control over design, and functionality. If you go that route, it is more important than ever to ensure the site is fast for all visitors, yet even more so for mobile visitors.
Too many sites even with responsive design, are toxic in how slow they are, at least intermittently, on various device types and across different connection types.
However AMP can be used for a site that is informational in nature. AMP can dramatically improve mobile speed vs. a site with bloated code and design.
If you go with two sites, you need to ensure both sites have the same internal links and the same core content, because the Mobile Index Google is working on, when it goes live, will use your mobile site or AMP site, to evaluate rankings on mobile devices. So if you fail to provide full navigation or full content, your mobile rankings will suffer.
Also be aware that while some publishers have seen tremendous growth after rolling out AMP, others have seen no value gain.
Google currently can`t fully and properly assign attribution of visits in Analytics consistently between AMP and desktop sites, though they say they are working on that.
The biggest unknown right now is inbound links. The concept is that if all or most of your links point to the desktop site and not the AMP /mobile site, your mobile rankings will likely suffer.
Except that is insane - if you earn high quality links that point to your desktop site, why should your AMP / mobile site suffer?
The Google team is still working to figure that out.
One final caution:
Google is pushing AMP right now. Yet they have pushed other technology in the past that they eventually abandoned.
Hash bang URLs, meant to help better process AJAX, was a big push by Google initially. But then they abandoned that entirely.
Rel publisher tags. Google Wave. Adobe Flash processing. Google Plus.
So sure, AMP may be a valid option, specifically for lightning fast mobile speed. Yet it may disappear entirely.
But if you go responsive and do not ensure fast speeds, your site will suffer in rankings for mobile more as Google takes page speeds even more seriously than they already have.
Gregory Dantschotter: Your pages (Mobile and desktop) have the authority value based on backlinks. So you don`t need to worry about Mobile and desktop.
Doc Sheldon: Ditto everything Alan said. I nearly always recommend responsive - NEVER m.
And I haven`t yet had a client for whom I felt AMP was the best choice. There are some instances that would call for AMP, I know... I just haven`t encountered one yet (and I hope my luck holds, in that regard).
But like virtually EVERYTHING in SEO, the real answer to your question is "it depends". ;)