In my opinion: SEO expert is someone works on the ways to build links, traffic and onsite and offsite optimization. SEO analyst is someone who analyzes the current seo of the existing site, works on keywords and target terms, niches and suggests what and how to do.
Expert to me, sounds like someone who can help optimize your sites on page and off page through backlinks. Analyst just sounds like somone who can evaluate your site and current backlinks and tell you what to do. Typically an expert will do more than just "analyze"