Hello, I have been lurking here for a while without speaking but the discussions are great and now you are talking about SEO too. :)
Link building is the biggest difference from regular SEO. Indeed, an SEO content strategist would probably not do that, especially as it is extremely time consuming. I also think SEO analysts do a lot more technical stuff someone concentrating on content will not normally do. As an SEO who in the past few years turned more and more to what I discovered was called content strategy, I see myself shifting away from URLs, indexing, rankings and link monitoring...
To me it is the difference between building something and constantly redecorating. Content strategy is building, which makes the combination of SEO and content strategy much more fun than spending a lot of time adjusting things (especially boring things like title tags and meta descriptions), nagging programmers and, worst of all, having to always depend on other people to come up with new stuff for the site.