RSS allows me to go directly to an article without having to navigate through the website. Some websites (TheVerge comes to mind) have a pretty nice design, but make it hard to get an overview of all the news and you easily miss things. With RSS I know that when I start my client all the articles they posted will be there and they will still be there when I start my client next month. I can manually mark articles that don't interest me as read, but keep others for when I have more time.
I follow about 50 sites, so if I had to check manually if any of them had updates (often in various sections, because they don't have a main overview that shows everything) I'd lose way more time. It's also synced over all my devices. I use the Reederapp in combination with Feedly, and it syncs over my Macs, iPhone and iPad. I have a backlog of about 500 articles, some months old, imagine having to go through a specific site and scrolling back page after page after page to get to that specific article you saw the title of 2 months ago.
I wish more sites would show the entire article in RSS feeds instead of a short preview. I understand the advertising reasons behind requiring users to click through, but with RSS you have a plain article that's optimized for reading without all the distractions of a website.