LastPass is a Cloud password program + browser plugin
So the extra layer is the plugin that is written by the same developer as the browser plugin. Which is a drop in plugin.
It's all open source so it's much easier for people to check the vulnerabilities. It's also easier to raise issues and other people to help fix them.
Separately Keepass is an offline database, so you have total control over access to it. The overhead of course is using something like Dropbox, plus probably Boxcryptor to ensure it's encrypted before it gets to Dropbox.
I've commented elsewhere here - it's similar to the Keepass bazaar vs the LastPass cathedral. Keepass might look uglier, but I trust it more.