The main limitation of static webpages is that you can't give different webpages to different people. But for many situations, such as a blog, this is perfectly okay. And the advantages are numerous:
- Fast page loads - you can't beat serving static files.
- Low server requirements - pretty much any system will do.
- No database - and therefore no security worries.
- No CMS - low learning curve and no heart-stopping moments of "The site is down!" (unless your server crashes).
- Easy to work with - just knowing html and css can make a beautiful site. With Jekyll, writing posts only involves knowing markdown, so anyone can do it.
Reading about advantages/disadvantages of static websites online will have you believe that all of them are unholy messes of unmaintainable orphaned .html files. But a good system like Jekyll can take care of everything that a CMS might.