Keep in mind the software you linked to is different from the software in the description you gave.
The link goes to LastPass
, the description is for KeePass
Lastpass stores your encrypted (supposedly) passwords in the cloud on their servers.
I trust KeePass more than Lastpass, since KeePass's database is only stored locally on your computer.
LastPass is easier to use because it will fill the login forms out for you and doesn't require as much user intervention. Supposedly they encrypt your database that's housed on their servers and they claim it's unreadable to anyone but you, but I really don't trust anyone that well.
If you want to make KeePass easier to use and automated, get the KeePassHTTP
plugin. Though it might be more advanced to set up that way. You need the Chrome or Firefox extension too.